IIRC that is a long slender firing pin with a furrule in the middle. the furrule whould have a flat on one side (filed in to it). If that furrule spins or is off center, that is what holds the thing correctly.
It sounds to me like you ARE putting it in correctly, so it fires. But the index is moving. If the part with the indexing flat is loose; a new pin or solder it might hold. Not a good winchester design, but figure they wanted an autoloader in their product line-up.
I'll think some more, but tough for me to recall the older guns, esp auto 22s.