If you can lay hands on a schematic that'd be a good start, it's no doubt floating around out there...
Okay, it looks like progress on the Applefritter thread, I second chasing that regulator is a place to start. You're lucky the Apple III uses proper regulators; I've been digging into restoring a TRS-80 Model I lately and the description of the power supply in the manual is enough to make a grown man cry.
(TL;DR: Radio Shack built it out of discrete components instead of using those 78/79xx series regulators; the rumor is they went that way because it was actually marginally cheaper to do it that way despite needing a larger PCB and at Tandy cheaper was *always* better. The way that machine gets its -5 volts is almost criminally shortcut-y but I guess it's barely good enough for 4116 DRAM chips to suffer through.)