15 amp adapter running the air conditioner

It's fine. Over 15amp is where the breaker may cut out. Typical RV air/con units run 9-12amps except on startup where they pull 2-3 times as much for a second or so while the compressor spins up to speed.

Breakers don't cut out instantly. The speed varies based on how much over the limit you are at 15.1amps, it may take a several minutes to pop. At 100amps it may take miliseconds. As long as it gets over the hump, the breaker is unlikely to pop. (of course, there is a bit of variability between identical breakers both out of the box and depending on how they have aged, so don't count on exact down to the 1/10th of an amp).

Assuming there is nothing else sketchy going on and the connection isn't getting hot, you are fine.

What was the voltage?

Obviously it is OK cause A/C Unit started/ran and no magic smoke got out OR did 15A Circuit Breaker trip (which is what would have happened if too much amperage had been pulled on that circuit)

It should be fine as long as the 15 amp outlet is wired properly and is in good condition. If it is an old outlet with corrosion in it, then it may overheat with the 12-13 amps worth of load that may be found from an RV air conditioner. This of course assumes there are no other loads present.