The easiest thing to do is to check the electrical connector to make sure it's tight. Since the catalytic converter was replaced, there's a reasonable chance the O2 sensor was removed and connecter was not plugged in tight when it was reinstalled.
You can check the sensor with a scanner, but since this is a '92 it probably doesn't work with OBDII scanners.
If you can take it out, you can inspect it to see if it's clogged or excessively sooty.
If you have a electrical voltmeter, you can also check it by removing it, put it in a vise, then heating it up with a propane torch. When it gets hot the voltage between the output terminals should be about 1.0 volts. When you take away the torch the voltage will quickly drop to zero.
Another way to check it is to leave it on the car but put straight pins through the wires so that you can hook up your meter while the sensor is in the engine. It should read about 0.6 volts with the car running.
One thing you mentioned that could indicate the O2 sensor is that the engine stabilizes after the CEL comes on. That could be happening b/c of the O2 sensor. The computer can run the engine "open loop" or "closed loop". Closed loop is when the computer is using the sensor reading to adjust air/fuel mixture. When it's open loop, the sensor output is ignored and the computer uses "default" settings.
The computer might be switching to open loop when the CEL comes on, thereby taking the O2 sensor out of the control loop.
A new sensor costs ~$75.
