Alternator it was 12.6 volts?

What's the question? It sounds like the generator has failed. One approach is to charge the battery at a slow rate for a few hours, then have the charging system tested.

Since these were redesigned for the 1987 model year, it is extremely common to have multiple repeat generator failures on GM vehicles. If yours has failed, to reduce the number of repeat failures, replace the battery at the same time, unless it is less than about three years old. That old battery will work fine in 1986 and older models.

This generator design develops huge, harmful voltage spikes that can damage the generator's internal diodes and voltage regulator, and can interfere with engine sensors. As the batteries age, lead gradually flakes off the plates, and they lose their ability to dampen and absorb those voltage spikes. A new battery will do a better job of that.

Let me know what testing shows up or how this gets solved.

There's nothing you can do with a bench test that you can't do on the engine. When working their hardest, it can take a good five horsepower to run a generator. The best you'll find on a bench tester at an auto parts store is a one horsepower motor, so they can't test a generator to its full output.

It takes about 20 seconds to connect a professional load tester, then, once the engine is started, all of the tests take just a few seconds; just long enough to observe the readings. That's far less time and aggravation than removing the generator for the less effective bench test.

The exception would be if it looks like your generator needs to be replaced. In that case, the people at the parts store can do the bench test to verify it's not working. This will show whether it's developing something or nothing, but not how much. The on-car test will show charging voltage, full-load output current, and "ripple" voltage.

This article:

https://www.2carpros.com/articles/how-to-check-a-car-alternator

shows how to start the test yourself. You already did this, but the article goes a little further. Normally we want to see between 13.75 and 14.75 volts at the battery, depending on which text book you believe, but you found it stays at 12.6 volts. On GM's 1986 and older generators, there was a test tab you could ground to determine if the voltage regulator was at fault. Those units were also quite easy to disassembly and repair.

The '87 and newer GM generators are a very frustrating design. Besides the voltage spike problem I described earlier, these are impossible to disassemble without destroying parts, namely, the thin, flimsy tabs on the diode block. At that point, you can't test the diodes. The voltage regulator can't be tested, and once apart, you can't test the integrity of the brushes. To be safe, you have to replace all of those parts, but that doesn't include the front and rear bearings which have been known to seize up on occasion. I like to repair down to the component level, but this is one time where a professionally-rebuilt unit is the best value.

By the way, when you go parts shopping, compare prices for replacement generators with a life-time warranty. My friend is on his sixth replacement generator on his '99 GMC. He found one with a two-year warranty for $200.00, and from the same store, one with a life-time warranty for $150.00. Better price and better warranty. He has had the lifetime one replaced for free three times.

There's one more "test" you should do before going any further. That is to observe the red "Battery" warning light on the dash. That bulb is part of the turn-on circuit for the voltage regulator inside the generator. If that light doesn't turn on when you turn on the ignition switch, we have to look at that circuit. If that bulb is burned out, there is also a resistor on the back of the instrument cluster to do the same job. In that case, the generator would be working fine, but the warning light wouldn't work. Also, if there's some other defect in that turn-on circuit, such as a cut wire or corroded connector terminal, generators will often "self-excite" from residual magnetism. Typically this occurs after the engine has been running for some time, and it's most likely to occur when engine speed is considerably higher than idle speed. When that happens, the charging system will continue to work normally until the engine is stopped. Even if that does occur, we don't ever want to rely on that to get the charging system up and running. We can discuss that circuit in more detail if it becomes necessary.