So, I've finally had the new roofer out to take a look at things. According to him, the reason why I'm getting leaks is obvious enough: basically, the roof was installed with a lot of corner-cutting and does not have the flashing it needs. (In fact, he says, the only reason I'm not getting a lot more leaks is because they installed the new pro panel roof over the existing shingles, which are still providing some protection.) Under other circumstances, I'd be a lot warier of "Oops, you don't need a small repair, you need something major installed!" as a verdict, but this, sad to say, is honestly about what I was expecting. Plus, they'd warranty the new work for another ten years, and, just based on a side-by-side comparison of the customer service levels I've seen so far, they seem infinitely more likely to actually make good than the original roofers, whose idea of fixing things appears to be solely limited to applying one more layer of caulk every time they come out. Not to mention the novel concept of actually communicating with me.
And in return, they want... Well, pretty much exactly every penny I could realistically scrape together right now without going too much deeper into debt. Sigh. But if it means I can fix this problem once and for all, and never have to deal with the other guys again, it's probably worth it.