1985 Camry alternator acting up in cold weather

So my 1985 Camry now has 144,000 miles and I only drive it about 5-7K per year.

In the past few weeks, always when the temperature is below freezing, I've experienced some rather unusual symptoms.

While traveling at high speed (65+) the horn will start blaring for no reason, and at the same time, the speedometer will go crazy, displaying a speed between 120-140 (the max speed on this cluster). It appears to be attempting to display a speed above 140, because the needle keeps jumping beyond 140 and bouncing back.

Slowing down to a speed of 20 or below solves the problem, at least temporarily. I do not have to shut off the engine or even shift into neutral. I just have to slow down and it all goes back to normal.

Once the engine has fully warmed up, the problems disappear completely and I can drive as fast as I want.

Then Tuesday morning, again it was cold, barely above 0, and I didn't get more than 1/5 of a mile up the street before I saw the battery and brake lights come on simultaneously, so I immediately thought I have a bad alternator.

I have a spare rebuilt OEM alternator that has barely been used since it was rebuilt by Toyota. Should I just put that in and hope the symptoms go away, or could there be other problems in addition to this? I haven't driven the car since the morning the battery and brake lights both came on at the same time.

I would put in the other alternator. That is the only way you can determine if there is another problem or not. May solve everything.

I think whatever it was had to be related to the cold. I changed nothing and drove nearly 200 miles on Saturday in a rain storm with temperatures in the high 40s and experienced no issues. I then drove 20 miles to work today after scraping ice off the windows and again I had no issues. I'm not complaining, but it's strange.