I may be wrong, but it sounds to me like the resistor that reads your key is damaged, or your key isn't providing the right resistance, causing the security light to go off and kill the engine for 15-20 minutes. My '97 Malibu used to do that from time to time, and I never did figure out why -- could have been moisture, gunk on the key, who the heck knows.
Supposedly, if you can figure out what resistance is supposed to be, you can bypass the key check entirely. It'* something inside the steering column. I looked up tutorials and diagrams online and debated doing it, but then I sold the Malibu to buy the Bonnie, so it'* someone else'* problem now.
Assuming that'* the problem, of course. I'm no expert on 88s. There'* no doubt someone here who'* more helpful than I am on this.