I have auto enginuity scan software on a laptop with the enhanced Ford package. Bring it by and I'll scan it for you and perform some other diagnostics. I need a problem car to practice using this software.
If it is not wanting to stay running when it is cold then it isn't going to be an o2 sensor IMO. The o2 sensor doesn't come into play until it enters closed loop mode. It's running off a preprogrammed mapping until the car warms up to operating temp. I'm thinking vacuum leak. How many miles? Ever done a tune up on it? It could be a coolant sensor maybe telling it the car is already warmed up when it's not and leaning it out.
Car has 250K on it. Did a tune up (plugs, coils, upper intake gaskets, fuel filter, air filter, oil change) about 15 or 20k ago. oil changes/air filter since.