Katie -
Here may be a couple things to consider. If any of these suggestions make sense, search the forum for that item/subject, and you should find a lot of data. Truth in advertising here - I have a 3.5 (2000), not a 3.8 on my car.
1. If I understand you right, you are saying that the engine starts OK when it is cold, and continues to run properly until it is turned off. By "until the gas runs out", I assume you mean "until the tank is empty". So you could go 300 miles without a problem, as long as the engine stays running. But that if you then try to start the engine when it is warm, it will start, but then will immediately stop (not stay running). And then if you let the engine cool, it will start and run properly again, until the hot engine is turned off.
2. FPR (fuel pressure regulator)? Fuel filter?
3. You said fuel pressure was good. Did you / can you put the pressure gauge on the Schrader valve (my 3.5 has one) and watch real-time what happens to fuel pressure when you are having starting/running problems?
4. I had an IAC (idle air control) cause my engine to fail to idle. However, it still started, as long as I held the throttle open.
5. There are a lot of posts about security circuit troubles. If your ignition system was "changed" (or do you mean "replaced" with identical, factory-stock parts?) the sky is the limit.
6. If it is a thermal problem, they are tough. But if it is, all the hot components continue to work, until it is time to restart the car. That would narrow down somewhat, which components it could be.
7. If the failure is reliable, it might be time for a shop/dealer to monitor a warm restart (since that is specifically when you have the trouble) with the famous Tech II diagnostics equipment.
Hope this helps...it's all I can think of.