there would be no problem adding a different wire, adding a fuse will add a bit more safety, as i say the factory wire isnt fuesed anyway
I'd start by poppign the plug out of the fusebox, see what the damage is. it maybe a previous owner has run too high wattage lights, and caused the wire to get too hot and melt the plug a bit. might just be tyou can get away with cleanign up the contacts and pinching the pin a tiny bit to make it grip more tightly.
if the fusebox looks damaged I'd recommend you replace it, who knows what else is wrong inside it!
Oh, and follow the wire back. some of the dash wiring passesd thru a massive black connector plug, sometimes the contacts in this get a bit poor, this could be the casue of your problem?
Once all fixed, I'd consider getting a relayed lighting loom. This will stop all the load from the headlights runnign thru the headlight & flasher switch, instead taking the load direct from the battery. this will increase the life of your stalk and light switch, plus give you better light output
