Really? Your shore power cord has a 15 amp plug on the end of it? I might be wrong, but I think that's got to be a modification by a previous owner. A 1988 would certainly have been set up for 30 amp, unless you had no original air conditioners.
Is the plug molded into the cord? If it is, I'd be inclined to think the PO changed out the entire 30 amp cord and swapped in a household-type cord. If the plug is just wired on, I'd think he just swapped plugs for some reason. You can buy new 30 amp plugs like this
You might want to check and make sure your wire sizes are adequate. They should be marked on the side of the cord somewhere. Your shore power cord should be 10-3, and your extension cord a 10 or 12. Almost all household cords are 14 or 16 gauge wire, which is OK for light loads but getting marginal for running the AC or microwave.
Check where the wiring is getting hot. Is it uniformly the entire cord set-up, or is it only or mostly at certain spots?
As I said previously, I would first check for loose or dirty connections, starting by pulling off the faceplate on the house outlet.