[ElStevano]

The problem is probably solved but not cured. It looks as thought the headlight pins will not connect fully to the female socket because it is indeed flexing. It's flexing a lot because half of the plastic wall/arm structure that holds the female socket in place is broken off. There is no way to push the two connections together by hand since you can't get at them in any way. (Although I admit I thought about trying to push the connection together with a metal rod through the small hole behind the side marker). Sooo, looks like I will need to replace the plastic 'sled' which the headlight capsule slides on. This part holds the female socket in place with an arm. I'll bet that will do the trick. I may just order the part and not go to the shop.

Thanks for the help in diagnosis, Toolpants.