The process is actually pretty straight forward and follows the official rules for CR calculation
First of all, sorry for posting an answer to such an old question, but after trying to puzzle out the Young Red Shadow Dragon's CR myself just now and seeing that none of the answers here actually calculate it the correct way, I felt the need to share my findings for anyone wanting to use that template in the future.
I admire the work the other answers* have done, such as inventing a Semi-Resistance category for calculating the effective HP or doing calculations on the Monster's experience points, but I want to tell everyone who wants to use the Shadow Dragon template and calculate its Challenge Rating: You do not have to do any of these. Calculating a Shadow Dragon's new CR, as exemplified with the Young Red Shadow Dragon in the Monster Manual, follows the official rules for determining a Monster's CR to a T, without needing anything else. These rules can be found in the DMG under the "Creating a Monster" Section starting on page 273.
As mentioned by you, a Shadow Dragon gains a lot of new abilities and gets changes to existing ones: Resistance to Necrotic damage, Expertise in Stealth, Necrotic Damage on its Bite and Breath and the fact that its breath now instantly kills and creates Shadows. However none of the above change its CR, the only ones that do are the resistances it gets through Living Shadow** and its newly acquired Shadow Stealth. The latter of which is listed on Page 281 of the DMG as one of the Monster Features that effects a creatures CR.
So to answer your question directly, to determine a Shadow Dragon's CR you just need to calculate the CR of the regular Dragon using the guidelines in the DMG but:
- Increase its effective AC by 4 because of Shadow Stealth
- Increase its effective HP because of the resistances from Living Shadow**
However, after that you will possibly have to recalculate the CR one more time, as these changes will often times increase the Dragon's CR so much that one or more of the following changes have happened:
- Its proficiency bonus has increased. This increases its attack modifier and saving throw DC and with that possibly its offensive CR
- Its CR is now over 10 and no longer requires you to add +2 to the effective AC for being able to fly and attack at range***. This decreases its effective AC and with that most likely its defensive CR
- Its CR has now crossed one of the thresholds for the Resistances Multiplier in the Table on Page 277 of the DMG. This decreases its effective HP and with that most likely its defensive CR
In the first calculation you will be using the regular Dragon's CR as the expected CR, while in the recalculation you will be using the CR you got as a result from the first calculation. I am certain this is the way they must have done it with the Young Red Shadow Dragon and therefore is the way we must do it, if we want to calculate the CR of a different Shadow Dragon for ourselves.
Calculating the Young Red Shadow Dragon
I am now going to demonstrate this process with the Young Red Shadow Dragon from the Monster Manual****. Carcer in his answer* tried as well to calculate the CR of that Dragon with the DMG's guidelines, ending up with the correct result CR of 13. However I believe this to be a a classic case of getting the right answer in the wrong way (as I am sure all of us have done in a Math Test at some point in our lives). The way I see it, there are at least five mistakes in his calculation, that I am going to correct in mine:
- He didn't consider the Saving Proficiencies
- He didn't adjust for being able to fly and deal damage at range***
- He didn't calculate the OCR and just guessed it to be 11 (which it is not)
- He didn't account for Shadow Stealth, as explained above
- He didn't recalculate in the end
First Calculation with expected CR 10
To calculate the total CR we need to calculate the offensive and defensive CRs first, called OCR and DCR for short. For the OCR we need its average damage over three rounds. Which will be using its Breath once (as it recharges on a 5-6) and hitting two targets with it and then on the next two rounds using its Multi Attack giving as a total we then have to divide by three to get the average per round. This results in this equation:
(56 * 2 [Breath hitting twice] + (17 + 3.5 [Bite] + (13 [Claw] * 2)) * 2) / 3
The result is 68.3 repeating or 68 rounded down. Giving us an initial OCR of 10, which does not change if we look at its Save DC of 17 as the regular Save DC for CR 10 is 16 and it only changes at a difference of 2 or more points. Note that we are using the Save DC here as more than half of its damage comes from its breath weapon.
With the OCR calculated we now look at its DCR, which we get by taking its 178 hit points and multiplying them by 1.5 for the resistances** giving us 267 effective HP and an initial DCR of 14. The DCR however is then increased further to 18, because the expected AC here is 18, while the Young Red Shadow Dragon's effective AC is 26. 18 from the actual AC + 2 for flying and dealing damage at range***, then + 2 for having four Saving Throw Proficiencies and finally the + 4 from Shadow Stealth.
So the first calculation gives us a OCR of 10 and a DCR of 18 which gives us a CR of 14 for the Young Red Shadow Dragon. We know this isn't correct, which is why we now have to do the recalculation.
Second Calculation with expected CR 14
For the second calculation the Damage Output does not change, still giving as an OCR of 10, however since its Proficiency Bonus increased we now have a Saving Throw DC of 18 (as seen in the finished Stat Block) which increases this OCR further to 11.
The DCR starts off with the 178 HP as before, but instead of multiplying by 1.5 for the resistances** we only multiply by 1.25, because of our new expected CR. This gives us only 222 effective HP and an initial DCR of 11. This is increased to a DCR of 14 however, because of our 24 effective AC, as opposed to the 17 which would be normal for a DCR 11. The Dragon's effective AC is now two points lower as it no longer includes the +2 Bonus for flying and doing damage at range, which is only for expected CRs of 10 or lower.***
So we end up with an OCR of 11 and a DCR of 14. These give us an average of 12.5 and its now up to the Statblock creator to decide whether it should be CR 12 or 13. WotC decided for the latter and I suspect this to be because of the Shadow Dragon's new Instakill-Shadowmaking-Breath, which as discussed doesn't directly change the CR, but can for sure be used as a tie breaker in such a scenario.
Calculating the Black Shadow Dragon Wyrmling
And now that we have shown the method works for the example Shadow Dragon Statblock from the Monster Manual, we can use it to create the CR for your Black Shadow Dragon Wyrmling.
First (and only) Calculation with expected CR 2
Its OCR is one Breath hitting twice + 2 bites over the first three rounds and then taking the average which is:
(22.5 * 2 [Breath] + (7.5 + 2.5 [Bite]) * 2) / 3
Resulting in 21.6 repeating which is rounded up to 22. This gives an initial OCR of 3, which is decreased to OCR 2, because of it only having a DC 11 for its breath.
Then its DCR is initially ½, from having 33 HP and doubling that to 66 because of the resistances**, but is then increased by 6(!) Stages to DCR 6 from having an effective AC of 25, which is 12 points greater than the expected AC of 13 for DCR ½. The effective AC comes from the 17 actual AC + 2 for flying with ranged attacks*** + 2 from having saving proficiencies and + 4 from Shadow Stealth.
So we end up with an OCR of 2, a DCR of 6 and thus an average, final CR of 4, for the Black Shadow Dragon Wyrmling. Which we do not have to recalculate, as its proficiency bonus has not changed and we also did not cross any of the HP thresholds.
To Summarize
Calculating a Shadow Dragon's challenge rating is a straight forward process that requires neither guesswork nor secret techniques, just the knowledge of how to calculate CRs in general. It just appears this conclusion might have successfully hidden from us for the last five years, like a Shadow Dragon in dim light.
*To clarify: My references to the other answers are not in any way meant to downplay these users or their work. The purpose of these references is to show that none of these are actually the right answer to the question, as none of them do the CR calculation the proper way, which is the reason I am posting an answer to this question 5 years after it has been asked.
**As pointed out correctly by Carcer it is safe to assume the Shadow Dragon will always be fought in dim light or darkness and therefore these resistances will need to be included in the CR calculation.
***As of now I am actually not entirely certain whether the CR of the Dragon's were calculated with the +2 AC from flying and dealing damage at range, which can be found under Step 14 on Page 279 of the DMG. Even though they literally can do this, they can only do so in one of the first three rounds of combat, and I do not know if this fits the requirement. I test calculated a few of the wyrmlings and their CRs where all correct whether or not this bonus was included. For the sake of consistency this post assumes that this AC bonus needs to be included. However I am open to anyone who knows the answer to this and might also research this topic further and adjust this post in accordance to what I learn.
****I could not have done this without the help of this Question that taught me how to correctly calculate the CR of the regular Young Red Dragon.