Rough cold start, cranks- won't fire when warm

I have a 98 Regency Elite w/ 3.8L non-supercharged motor, 288,000 miles, put 241,000 on it, and still like the car.

It starts and runs REAL rough in cold weather and the check engine light comes on, but if I shut it off and immediately re-start it, it magicaly runs great.

After it's running, driven a while, and up to temp, it won't re-start. It cranks very well, and seems like maybe one cylinder fires once, but after that, it just cranks with no more firing. It eventually starts after repeated tries (maybe 20-30 tries, I wasn't counting), just bumping the starter briefly, without touching the gas pedal. After it finally starts, it runs perfectly. ?? As a side note, I just replaced the plugs, wires, and coils because of wet weather issues.

Any clues for this starting mystery would be appreciated.

Welcome to AF, first get the code, also check fuel presure when it will not start, and spark at the plugs, basic initial diagnostics