There have been several threads on 796 headlight wiring, some of which I have contributed to. My final solution to the bad white connector was to eliminate it and wire in an extra length of similar gauge wire to allow for bulb changes. That was after the dealership botched the repair under warranty at that time. I took extra care to get good solid connections because the frame side available wiring was so short. Now my final solution has overheated and burned and it's going to be a PITA to fix, maybe gas tank and airbox off to get access to the loom. Before I start, any ideas why these wires are getting so hot? How do I prevent the same thing happening again? I thought there was another thread on that issue a while back but I just can't find it.
Sounds like you need heavier gage wires. And maybe a relay.