I've tried equipping an alchemy poison, but the only thing that happens is that I drink it & get poisoned. How do you apply it to your weapons?

You can't. sadly.

edit: if you're on PC, however, there is an interesting mod that you might want to check out. (assuming you want to use alchemy for combat)

http://mw.modhistory.com/download-98-10001

I tried it years ago but decided not to use it. Unfortunately I can't remember why.

In Morrowind the only uses for "poison" potions is getting your alchemy skill up, and selling.