1993 Acura Integra distributor?

Every once in a while over the past year after the car has been driven a half hour or more, I would stop somewhere and come out 5 or 10 min. Later and the car would crank, but not start. I would wait a couple of minutes and it would start. When summer started up the problem kept occurring more often. At one point, the car started idling real low and I discovered that the rotor button screw lying in the bottom of the distributor cap. I replaced both because there was visible damage to both, but everything else looks fine. A couple of weeks later the crank/ not start problem started happening often. I checked for spark and got none. The next day the car wouldn't start. I followed the steps in my guide book. I check the resistance on the ignition coil and checked to see if it got 12volts when key was on. Everything checked out good. So the book said to replace the Ignition Control Module. I did that and tried starting the car 3 times and got crank but no start. Went inside and got somebody so I could check for spark. The car started on first try. I tried starting the car around 12 times and it started right up everytime. I then tried starting the car a half hour later 5 times and the started every time. I knew I had a gallon or less of gas in the car and thought maybe thats why it didn't start after I installed the ICM. I drove 15 mins to work the next day and everything went fine. After work it took about 5 tries before the car started, again I thought maybe low gas. Drove 2 min. To the gas station. Filled my car up and learned I was down to about a half a gallon. Then car would crank and not start. I tried and tried over a 10 min period. I check for spark and got none. I tried again and again and eventually it started right before I was going to take off the distributor cap. I drove home and noticed that the car was idling lower. At home kept trying to start the car. It would start about half the time. I got no spark the times it didn't. I talked to my moms friend who is a mechanic and he said I got a bad ICM and to exchange it for another. Pepboys didnt have another one and after talking to them they were positive that the distributor is causing the ICM to overheat and malfunction. My moms friend says that that doesn't happen and they are trying to get me to but a distributor. So can a distributor cause the ICM to overheat, or is it a bad part, or maybe something else all together?