You said the car has a "relatively new" transmission, I suppose the first thing to do is check the transmission fluid level to ensure it is correct if not top top up with the Dodge approved type transmission fluid. If a Check Engine/Service Engine soon light comes on a code will be stored in the powertrain computer in your car that is retrieved by using a scan tool connected to (a common to all 1996 onward) a 16 pin diagnostic socket under the dash of the car. The code stored will indicate if there is a solenoid, toque convector or other transmission related problem. In short you will need to bring your car to a transmission specialty shop or repair shop with Dodge/Chrysler experience. Get the car checked out properly or a poorly running engine will damage a catalytic converter over time.