Turning over but will not start

[dlwdkh]

93 Grand Am with 3.3 litre V-6 after driving it short distances and turning it off it will not start back up. Have two of the things that have nearly same problem. One will turn over but will not start for usually about an hour. it has spark and the fuel pump is pressurizing. after waiting it will start and run fine. If it gets heated up good it it will start fine after no time at all, but if it doesnt we have to wait.
The second one with same motor and same year does something just a bit different. If you drive it for a short distance and shut it off it wont turn over. The solenoid does not engage. Hit the key and no click no crank, no nothing just like the battery was unhooked. It is not the solenoid though or the starter. Have taken these and had them checked out thoroughly by a starter and alternator shop, twice.
Both start and run fine after cooling back to around the air temperature. I suspect it is electrical and something to do with temperature and the computer shutting down something because of the temp. but have no idea how to check it out. any one out there have any suggestions or the same or similar problems.

[Brittanne]

Similar problem with 91 Grand Am... could start amazingly well in cold conditions, but, after turning it off once, it would have trouble firing again (but lights, radio, etc. would come on). It was my crankshaft position sensor.