1999 Sebring JXI Convterable - Won't start. No panel lights when key is turned.

My problem started off when car was warming up in driveway. Came out and it had shut off. When I went to restart, I got nothing. No dash lights except odometer lites up  and Tachometer Needle moves. I let it sit and when I tried a few hours later, it worked fine. Then it did again. And again. Only after it sat idling for more than a few minutes. Each time after letting it sit, it would work fine and restart. Had a couple shops look at it and $700 later, it is still doing it. Worse now. Now when I shut it off, the problem will rear it's ugly head. AND... Before I could let it sit a half hour and it would restart. Now it has to sit a day or more. I put a Brand New Battery in it yesterday and the problem still persists. Tried to start it this morning and nothing YET! I have spent 3 hours reading questions on this site and could not find the same problem. Tried ideas from somewhat similar problems, and none of them seem to work. Help.