My 1993 240 wagon has had a cold starting issue for about 6 months now, but it's actually getting bad enough that I'm trying to solve it now... any ideas would very welcome.

On cold start (cold engine, not weather-related. it's been doing this every day since July) it seems to run fine for a second, then immediately gets REALLY rough, missing, etc. If it can maintain idle, it usually will clear up in 30 sec - 1 minute or so... but sometimes, it dies and needs to be cranked for quite a while. Definitely flooding, the plugs are wet...
If it does clear up and start, it then runs completely fine. No issues with idle at lights, etc. It also is fine with hot starts (if it's been running recently).

I haven't been paying close attention to mileage, but I think it's worse than it used to be, probably always running somewhat rich.

FPR and FI relay seem fine, but I don't think that those would cause it to run rich.
I gave it a tuneup a while back - new plugs, cap/rotor, wires - with no effect.
No check engine light, but when I checked the OBD, got a code saying O2 sensor is faulty/missing. Am I correct in thinking that the O2 sensor should have absolutely no effect on cold start issues?

I just swapped in a spare AMM to see if that has any effect, should know if it does tomorrow AM.

My next plans are to test coolant temp sensor or cold start injector - any suggestions for quick testing of these? is there anything else that I should be thinking of?

Thanks for any assistance!