2001 Buick Regal starter problems

2001 Buick Regal mileage: 44,000. I am having starter problems. The car will run fine until it warms up. If we stop and shut the engine off and try to start, it up 10-20 minutes later, all I get is the starter clicking. I have to let the motor cool down for several hours before it will start again. I have had the battery checked and the alternator both tested fine. I have to check the fans to see if they are coming on and check the contacts on the battery. There is something ever stranger. When it does fail to start, I have noticed that the clock on the radio is behind an hour or two. No idea what is causing this. I am leaning toward the starter and starter solenoid at this point. Any other ideas at this point would be greatly appreciated. I want to try to fix this myself before I take it to the dealer.