Update: After posting my problem, I read through a lot of the prior postings. This might be a problem people are solving in different ways.
Bottom Line: I replaced the neutral safety switch located in the console and activated by the automatic transmission shifter. I will repost when I am certain this fixed the problem, but all looks good at this point.
What is Happening: When the car gets hot, the starter, solenoid, wires, etc. get hot ant it take more current to engage the starter. You could get around the weak neutral safety switch by letting the car cool, shielding the starter, insulating wires or moving them away from heat, or you could have a bad or weak ignition switch.
What I Did: I ran the car until it would not start. I ran a wire from the ignition side of the solenoid (cliped by an alegator
clip) to a cheap 12 volt tester (alegator clip with light and a sharp point). I grounded the tester to a screw on the sill plate while trying to start the car. Although the light came on, I don't think the starter was getting enough current. I started the car by touching a screw driver across the positive connection on the starter to the solenoid ignition wire connection (the one and only small wire on 89 camaro solenoid). I went to a parts store to replace the starter switch or neutral safety swith, but the parts man could not tell me how to remove the starter switch, so I went for the neutral safety switch.
Thanks and Keep The Ideas Comming: All your comments have been a great help to me. I was about to give up on this problem and take it to a dealer who I know would charge me over $200, keep the car for three days minimum, and still might not fix the problem. Even if I am on the wrong track, you have given me more ideas to try and think about. Until I am certain my problem is solved, I appreciate additional comments.
As someone else said, Prayer always helps too.
Hints For Replacing Neutral Safety Switch: First disconnect the negative wire to the battery. Put the emergency break on. Remove the C clip from the shifter knob. Hit the bottom of the knob with the open part of an opened end wrench or something that will go around the shifter post with an upward motion. Don't scratch the post and watch out the knob might pop up high. I had to remove/loosen the screws holding the shifter to remove the old switch. When I pulled my switch, I noticed it would move from side to side, but the new switch did not. The parts man said that the switch should not move unless it is engaged (i.e. has current, shifter in correct position, etc.). I had to turn the key and depress the shifter (where knob would press down) to get the post aligned so I could install the neutral safety switch. Put the car in Park and reconnected negative batery cable. Started car.