Might want to check the ignition relay. The ignition relay is on the top side of the under dash fuse block, can get access by removing either the storage bin and/or air vent/mirror/dimmer control trim piece. This
diagram is from another thread, the red arrow points to the
starter relay, but you want to check the IG1 relay. I think its the same relay used for power windows (P/W relay in the
diagram), so an easy thing would be to switch the P/W and IG1 relays, see what happens. Might not fix it, but an easy thing to try. (might want to first make sure your power windows are all the way up, in case it does work)
here's the
diagram:
