I can only see a few things that could cause your problem.
o If a slight wiggling or lifting of the stalk returns low beam function that would indicate the low beam contacts in the turn signal stalk switch, however you seem to have to try harder than that. The main headlight switch shouldn't be at fault or you wouldn't have any front beams.
o That pretty much leaves a relay problem. There's a headlight relay (square, should be front row, second from left) and the bulb sensor relay (round, should be back row, leftmost) involved in the low beam circuit. Lift up those relays a bit and reseat them to ensure good contact. One big sign of poor contact would be darkened or melted plastic around the contact sockets, especially the headlight relay.
o If letting things cool down seems to help then suspect that the headlight relay is internally overheating when the high beams are on. If tapping the headlight relay while it's malfunctioning returns low beams then that would also confirm a headlight relay problem. In that case you could try resoldering the main internal relay contacts, but a replacement (used is probably okay) may be required.
o There could also be a bad low beam circuit in the bulb sensor relay. Again you could try resoldering the internal contacts (a bit more advanced), but a replacement may be required (used is definitely more affordable).
I strongly suspect you'll find it's the headlight relay. I have to ask, does your car have high wattage headlamps? If so they may draw too much currrent for the existing electricals. That could easily fry a standard relay or overheat wiring or melt a headlamp connector.
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Dave -not to be confused with a real expert, just goofing around at this
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