How can I get through very long and very dry, but also very useful technical documents when learning a new tool?
Man, I'm right there with you. I understand completely your situation, as I've been there myself many times.

The key, I find, is not reading the tutorials. 

Instead, I pick a few projects - starting with something that's "hello, world" level. I google and read enough to be able to do that. Then I pick a slightly harder project - for Vulkan, maybe pick a scene of intermediate difficulty, and google and read enough to do that.

If you keep doing this, you very quickly obtain a practical base of knowledge. There are, however, disadvantages to this - obviously, there will be holes in your knowledge, and you could miss style guidelines depending on what you're looking at to write your code (e.g., when I first learned python, I came into it knowing how to program in javascript - so my first instinct was to write variable names in camelCase, and it was only after several years that I actually read PEP8 and changed to under_scores, though in my defense I didn't even realize style guides were a thing at the time). 

You might even wish to go through this approach, and then read through the tutorials with the perspective of someone who's "brushing up" - this might fill in the crevices the more trial and error method doesn't fill.

If, however, you wish to fully work through the documentation straight off, I'd recommend reading a paragraph/section, and writing down something about how you'd use what you just learned about in a real program - i.e., mentally applying the small details. You want to build up a 'web' of details, with lots of connections between the nodes, so that every node is much easier to 'get to' (remember). The more you know, the more you remember (as odd/obvious as that sounds).