I had that problem with my Stratus. Never got it fixed.
Is yours a coupe, sedan or convertible? If it's a sedan or convertible, there's a "calibration" technique you can run.
With the engine running, turn the temp control all the way to the cold side.
Turn the mode control all the way to the defrost position.
Put the fan switch to any position other than off. Then press the rear defroster button until you hear a chime.
Let it go. The system will go thru some gyrations and give you a "report" in your odometer readout.
A "0" is good.
You may have a bad control head. As you noted, the default mode is defrost, so if the body computer senses a loss of signal from the controls, it will put it into defrost.
Again assuming yours is a convertible or sedan, salvage control heads are fairly commonly available on eBay and in junkyards.
The part is used in the sedan and convertible and in the Dodge Stratus from 2001 through 2006, so there are lots of them out there.
It's an easy replacement which I would definitely try before tearing the dash apart.
The IOD fuse is the one in the middle of the fuse panel at the left end of the dashboard. Easily recognized by the "ears". You can pop it out with no tools.