I have this problem on my 91 TA vert (350 tpi) where after Ive been running it for atleast 5 minutes or so, if I turn it off it wont start. Now if I goto start it up within about 30 seconds of turning it off, it starts right up, if I wait anylonger, I have to wait a good 20 minutes before itll start. Its a stock 350 TPI with only mods being SLP Y pipe with random tech. cats, SLP headers, Hooker aero chamber cb (3" all the way back) the Headers are very close to the starter. Could this have something to do with it?

My father thinks its the selonid cause when you hit the key, you dont even hear the selonid energize.

 

If there is no starter activity after heat soak, your father may be right.