What should I do if my old Apple Watch won't charge?

If the question is what to do, then you would best get a new watch, that one is obsolete and cannot be repaired.

I see that you have a Series 1. Running watchOS 6. This is a relatively new software for the Series 1 to be running and it could run into issues with that. (Out-of-date processor and all)

The best thing you can do is what basically everyone will tell you. Buy a new one.

At this point, the AppleCare on that thing (if there ever was) is long expired and can't be serviced for a battery replacement. So, your only option is to buy a new one.

Some recommendations on newer watch models to buy:

Series 10

Ultra 2 (if you have the money and if your wrist is huge)

Refurbished Series 9/8 (still pretty good today, just might run a little slower)

SE (about as good as the Series 6 and still works well for its lower price point)