I just did a quick calculation - based on your suggestion - with linear demand/supply curves and what I got as the aggregate price is a weird combination of the two prices. It's not any kind of (weighted) average of the two, as far I can tell. In particular, it seems to me that the aggregate price will differ if, for one of the markets, we change either demand curve or supply curve but keep their intersection at the same point. That's wildly counter-intuitive, I don't know why should aggregate price behave this way.
Maybe I am wrong, and you can make it work. If you do, write it up! It really should be in textbooks, it's embarrassing they don't discuss this case.
And this is a case of two (presumably) completely independent markets with linear curves. It gets much more hazier when you have a relationship between the two markets (for example, one is a substitute for the other).
Disclaimer: I've never even taken Econ 101, so I might completely misunderstand something. If that's the case, please downvote and tell me where I'm wrong.