Two and a half hours later (some of the time spent running to two different neighbors looking for a pipe wrench big enough for the job) I have clear pipes.
I took the little "U" shaped plastic pipe off, ran baking soda and vinegar with a chaser of boiling water out the pipe that comes straight off the sinks. Thinking that all was clear, I put the plastic pipe back in place, ran some water, and immediately had plugged up sinks again. Which of course meant that the plug was in the pipes that lead into the wall. Off with the plastic pipe, again.
Luckily, I recently learned how to use our plumber's snake (in the bathroom, another unpleasant but ultimately successful adventure in plumbing) so I ran it up the pipe. Four times. Each time I pulled it out, I had cucumber peels and potato peels embedded in the head. Gross. I brought the snake out to the front lawn to clean it out (with a shish kebab stick, no less) each time, and each time I had a little less vegetable matter to poke out of the coils. I'm sure the neighbors really wonder about me now, squatting on the front lawn in my rubber boots, picking at a plumber's snake.
After the fourth snaking I decided I'd had enough and wasn't pulling out that much anymore anyway. Put the plastic "U" shaped pipe back on (I'm pretty certain that is not the official term for that piece of pipe, but there it is), put more baking soda and vinegar down the drains, panicked a bit when there occurred a little back-up. But with some seriously hot, fast flowing water, the plug finally burst free and now I have clean pipes.
Saved us probably a couple hundred dollars and have earned my plumber's bragging rights for today at least.
I've learned two valuable lessons here:
1) if using the garbage disposal, I must run water for a lot longer than I have been to make sure everything is flushed away;
2) this is probably a sign that instead of using the garbage disposal I should try to resurrect my compost pile and put all the veggies in there, instead.