From what I can tell, it's your "cross-reference and educated guess" scenario.
I don't own any of the modules myself, but this blog post details the varying availability of dates in published materials:
Some products bear no dates at all. Others hide a date in some hard-to-find place, or the date can be inferred. Some products have a clearly stated date. And some products have incorrect date references!
They conjecture that this is likely intentional, to create and address the continuity as a whole but also give individual DMs great leeway in making it their own:
D&D’s Conflicted View on Canonical Timelines
The 5E D&D team struggles with canon for good reasons. They want the world to feel deep and real, and to have events matter. Simultaneously, they don’t want us to pick up a product and feel like it is out of date, or that we must play it in a certain order. The world should feel accessible and flexible, easily fitting a DM’s needs.
5E products have varying approaches to balancing these conflicting needs. [...] In several products, designers can’t help but include fun nods to previous adventures. Those elements enrich the adventures, at the cost of forcing a timeline.
DMs and players that want verisimilitude find themselves hunting for dates. That’s a big part behind why this blog post exists: to give us that timeline we often desire. However, don’t forget the reasons why Wizards hides the dates. Our home campaigns can do whatever we want with these dates. We can skip adventures. We can play them out of order. The world is truly ours.
They proceed to go year by year, describing their evidence and reasoning for what happens when, such as this example for 1491 DR - and, naturally, there are gaps and guesses:
1491 DR – Princes of the Apocalypse adventure takes place. The novel Death Masks takes place, and those events reference the adventure Curse of Strahd, which therefore takes place in 1491 or slightly earlier. Lost Mine of Phandelver, from the Starter Set, likely takes place here (probably before Princes). Pages 30-31 of Lost Mine of Phandelver describe the eruption of Mount Hotenow (1451 DR) as occurring “30 years ago”, which would place the adventure in 1481 DR. However, the Acquisitions Incorporated book suggests Phandelver and Princes take place at the same time (and Princes is stated as taking place in 1491 DR). Ed Greenwood said on Twitter, “Lost Mine is officially a bit nebulous in date for DM convenience, though there are detailed internal WotC timelines so as to keep things straight (i.e. novels). Safe to say: 1490s DR.” The date of the eruption is incorrect in another 5E product, so 1491 is likely the correct date for LMoP. Tomb of Annihilation likely takes place in either late 1490 or 1491 (or very early in 1492), as it features frost giants searching for Artus Cimber and the ring of winter. Both Storm King and Tomb name Jarl Storvald and feature the frost giant captain Drufi with the same accompanying bodyguards and beasts. Drufi leads the assault on Bryn Shander, and it is noted in Storm King that if Drufi does not find Artus there, Jarl Storvald sends Drufi back out in search of the ring (and perhaps to Chult). Since Storm King has variable starts and not all locations are visited or all villains defeated, this is plausible. Tomb has Volo writing his book, Volo’s Guide to Monsters, and Waterdeep: Dragon Heist mentions the book is published, so Tomb must take place before Dragon Heist.
So... yeah, good luck.
