This is all so helpful everyone! It seems like there are three major functional changes that I may want to make, and I think both of them could help with getting leverage on the brakes. One is changing to dual pivot brakes, one is adding interceptor levers, but another that we haven't talked about here yet is raising the bars. The bars can't go any higher (or at least not much higher) on the current stem. To my eye, they aren't quite up at the level of the saddle yet. I can definitely reach the hoods, but I'd like the resting position there to feel a little less stretched out. I figure if they could be a bit higher, and if the stem were maybe even a little shorter, I might have more leverage on the levers.
What do you all think? And does it make sense to change the brakes first or the bar height/distance? I'm hesitant to mess with changing the stem until I've gotten a little more used to drop bars, because maybe I'll just get more used to stretching out. It hasn't been hurting my neck or back, though I definitely feel like I'm still adjusting. On the other hand, I don't know if it makes sense to change the brakes until after I've tried out the current ones with the different bar position.
The interceptor levers seem like a separate question, just as a matter of personal preference whether I want that other option. Either way I want to be able to brake comfortably from the hoods.