Just to be sure, montokboy, you do know that when the key is off or in the accessory position the healights won't come on but the parking lights will. The headlights only turn on when the key is in the engine running position. I'm sure you know that but sometimes you gotta make sure.
Your wiring should not be affected by heat. There are only simple relays and wire, none of which should be affected much by heat. Power for the headlights does not go through the headlight switch but power for the relays that activate the headlight circuits does.
Try this. Turn your headlights on (with the key in the engine running position, of course) and try tapping on the K3 and K4 relays. They are the only relays with contacts in series with the headlights that could affect their operation. If you're careful you can even take these relays apart, at least the metal ones, and see if the relay contacts are clean. If the contacts are pitted and corroded you can clean them up and see if they work better.
I would suspect the low beam relay before the high beam relay but both can cause the problem you're seeing with the low beams. However the low beam relay is very much more the likely culprit.
Another thing that could cause it is the ground wire on the headlight switch. If that one's dodgy it would screw things up as well.
Good luck.
Steve