This is a very well known issue with the 03 and some 04 cars. The ductwork in the seat becomes crushed as you sit, which blocks air flow when the seat climate control system is on. The system responds with an overheat or overcool error code and shuts down. It wont reset until you shut the car off and restart. The diagnostic trouble codes (DTCs) for the shutdown are stored in the car's memory, so the techs will see these when the run a diagnostic. In addition, there is a Technical Service Bulletin (TSB) on this at the Lincoln dealer, so the techs are aware of this problem. Tell them to refer to TSB 04-18-16. The fix is basically to take the seat apart, fix the crushed ductwork, and reassemble the seat.
Is your car still under warranty? If not, I'm not sure what this will cost you, but I don't think it will be cheap (lots of labor). When this happened in my 03, I got it fixed without charge under warranty.