I'm guessing you removed the sync chain when you cleaned the bike? If so, and if the chain has been on there for a long time then the quick fix to eliminate the pitch difference between the chain and rings is to advance the chain by a single tooth.
Is this wear going to give us problems on RAGBRAI? So long as you get the original chain back in the original orientation / synchronization with the timing rings and can adjust out any slack in the sync chain it will work just fine.
Shall we switch to the right pedal when we start? Doesn't matter....
Can I remove the chainrings and time them so the worn teeth are not engaged when we start? Actually, unless someone else has done this once before, you can swap a set of well-worn timing rings from front back / back to front and have, in essence, a brand new set of timing rings. However, you'll need a new sync chain (takes 1.5 standard chains in most cases) as the old chain will have stretched too much to work on the fresh teeth with correct pitch.
If I need new chainrings, am I going to have a hard time finding them? It is a 1996 Shimano XT 8-speed. It shouldn't be too hard; they're basically middle / small chain rings. You just need to match the 4 or 5 arm configuration and bolt circle diameter with the replacement rings. You can use non-ramped and pinned rings or rings with ramps and pins; it doesn't matter. You can also use pretty much any matched pair of chain rings using anywhere from 34t - 42t; 38t - 42t are the norm.