Thanks for all the ideas!
Someone had told me a similar theory about the valves, but maybe someone can educate me - I was starting to think maybe that wasn't the issue, because I'd think in that case the oil would burn off within a minute or two? I was trying to go up a very long hill, and pushed it a little bit to see if the oil would burn off and it didn't seem like it was getting any better. There's so much smoke that it seems like something very bad is happening, so I've been reluctant to push it too much. But if it is just oil being burned off then I'm guessing it's not causing any more "damage" than already exists?
I've heard mixed reviews about Bar's Leak but may be getting desperate enough to give it a go.