For the next guy with this kind of trouble, I found the problem.
After an entire weekend with my multimeter, it occured to me that the only thing the turn signals, radio, whipers, and probably a couple of other things as well is the ignition switch.
When you turn the key, current to those things are interrupted while starting and returned when the key turns back to the run position. Wear, grease, and corrosion inside the switch was the culprit. I took the switch apart (not for the faint of heart), cleaned all the lithium grease out (why lithium greas inside a switch?!), cleaned the contacts everywhere, and put on dielectric grease instead. Works like a champ again.