My Corolla's battery light won't turn off. Any idea why?

I have a ‘94 Toyota Corolla. The battery light came on this morning. Cranked just fine.

Tonight I went to Jimmy John’s for a #10 (I usually get a #9), and when I got out to the car, it wouldn’t start.

Grandpops came out and jumped it for me. I got less than 50 yards up the road and it sputtered and died (shortly after all electronics turned off.

I just got a new battery and put it in, replaced the (-) terminal connector and the (+) one looks in good working order. The car is now running strong.

BUT!!!!!!!! The battery light will not turn off now.

Any ideas?

If you say “the alternator,” can you also mention where I can get that diagnosed without having to pay $80 for them to tell me “it’s fine?”

Any suggestions are much appreciated.