[Richard]

So, resolder the cold joint which has given high resistance leading to too much heat.

Then clean up the existing connector.

Put it back together and see what you got.

Worst case you can get 1/8 female terminals, cover them with heat shrink, and manually connect it.

You are right on the connector. You can buy them from places like Mouser. You will need the internal pins and the tool for crimping/inserting them into the connector.

I had the same thing happen to me at the same connector.