Your calculation for the resistors needed is incorrect.
To limit the current in an LED string you would need to know how much voltage there is across the string, how much voltage there is from the supply and how much current the string takes.
The resistor needs to drop the difference between the supply voltage and the LED voltage at the LED current. To be able to regulate current you need to have a higher supply voltage than the required load voltage in order to have some headroom. For example to get 36 V at the load you might use a 40 V supply, to figure the resistance needed you would use the difference in voltages:
$$R = \frac{40V-36V}{0.35A} = 11.428\Omega$$
And the power dissipated in the resistor would be:
$$ P = (0.35A)^2\times11.428\Omega = 1.4W $$
The value you calculated was the power the LED strings would dissipate, \$36V\times0.35A=12.6W\$ per string.
A resistor by itself won't actually 'limit' current to a specific value unless the resistance is a significant portion of the total load.
Let's look at what happens if we use a 40 V supply and 12\$\Omega\$ resistors. We'll look at the current for 3 strings, one with the specced 36 V drop, and two that are a bit off, 35 V and 37 V. Ideally you want the resistor to equalize the current through each string so they have the same brightness.
@35 V \$I = (40V-35V)/12\Omega = 416mA\$
@36 V \$I = (40V-36V)/12\Omega = 333mA\$
@37 V \$I = (40V-37V)/12\Omega = 250mA\$
Hardly the results we want. This is because the LED strip impedance is around 103\$\Omega\$ so 12\$\Omega\$ is a small fraction of the total.
If we upped the resistor values to 120\$\Omega\$ it will be more than half the load, but at 350mA they're going to drop much more voltage, around 42 V. That makes the required supply voltage \$36V + 42V = 78V\$.
Using that in our previous calculations we get:
@35 V \$I = (78V-35V)/120\Omega = 358mA\$
@36 V \$I = (78V-36V)/120\Omega = 350mA\$
@37 V \$I = (78V-37V)/120\Omega = 341mA\$
This is much better current regulation, but at the expense of a much higher supply voltage and ~15 W dissipation per resistor.
Resistor current balancing for LEDs works pretty well when dealing with single low current LEDs such as in a 7 segment display, where the LED impedance is around 70\$\Omega\$ and you use 180\$\Omega\$ resistors @ 20mA from a 5 V supply. For high current LED strings, not so much.
A better method than a simple resistor would be a current regulator or constant current source (CCS), this can be made with a couple of transistors, or for more control an opamp reading the voltage across a sense resistor and driving a transistor. You can see this Wikipedia page to get some ideas of how to do that.
Here's a simulation of a basic example of a current limiting circuit just to give an idea of how it works. I'm not guaranteeing it will work in practice without some modification.
The voltage across each 5.6\$\Omega\$ resistor is held constant by the voltage reference at approximately \$2.5V - V_{be}\$. Since the voltage and resistance are constant the current must be constant too to satisfy Ohm's Law. The voltage sources V2 to V4 represent 3 LED strips with the maximum range of voltage drop from the datasheet. The supply voltage is swept from 44 V to 52 V. You can see that the currents stay pretty close to 350 mA. 48 V supplies are common, so one of those could be used to power it. The transistors would need to be on appropriate heat sinks.
