Battery light stays on?

Start by taking some voltage readings. First, see what you find on the fat, bolted-on output wire. If possible, I'd very much prefer you check this with a test light with the ignition switch off. The light should be full brightness.

Next, measure the voltage on that output wire and on the two smaller wires with the engine running. These are only valid with the engine running, and with everything plugged in and connected. That means for the two smaller wires, you must back-probe them by sticking the meter's probe through the rubber weather seals alongside each wire.

Let me know what you find with those four readings. In case you aren't familiar with using a test light or reading a digital voltmeter, start with these articles:

https://www.2carpros.com/articles/how-to-use-a-test-light-circuit-tester

https://www.2carpros.com/articles/how-to-use-a-voltmeter

You can find both tools at Harbor Freight Tools for $7.00 or less. Also look at Walmart or any hardware store. In the article, they're using an "auto-ranging" voltmeter. That's an expensive option you don't need. If you need help setting up or reading your voltmeter, I can help with that.

P0622 - Generator Field "F" Control Circuit Malfunction

I need voltage readings to analyze. What did you get for the four readings I asked you to check?

Have you made any progress? What do you mean by, "everything checks out", and by "nothing"? No problems found? It's not working? I'm here just about every day, but only for a few hours, so don't panic if I don't respond to your reply right away. This should be an easy problem to solve, once I know those voltages.

The last one like this found some voltage on the ground wire indicating there was a break in that wire, or the connection was rusty where it bolted to the body sheet metal. Once we learned the solution, that person never replied back to say where they found the break. Regardless, let me know how you're doing or if you solve this.