I am not 100% certain on the 2002 model year E350, but....
For model year 2007, that special parking brake is attached to the drive shaft directly behind the transmission, but ONLY THE E450 cutaway has it. The E350 cutaway has rear disk brakes like many cars with the parking brake being a miniature drum brake inside the hat portion of the brake rotor. Any repairs to it should cost a little more than a car. A $2200 quote is a snow job.
Driving 5 miles in town should not ruin the parking brake. A worst case scenario would be that the drum portion and miniature brake shoes formed a glaze which could easily be cleaned off using sand paper to provide better gripping power. A mechanic would have to remove the brake rotors which involves a cotter pin and wheel bearings, but that should cost around 2 hours of labor, not 20 hours of labor, hence $2200 is a rip off.
If doing that, I advise to pay a little extra to have the rear grease seals replaced and the rear wheel bearings packed with more grease. Grease seals might run $10 a piece. A mechanic with integrity might charge an extra $50 to do that for both wheels.
As the previous reply suggested, I wonder if the parking brake cable mechanism needs lubricating, maybe also adjusting so the parking brake pedal is more effective.
Doing a quick search for clarification, I found this diagram of a miniature drum-&-shoe parking brake system inside rear disk brakes for a typical car. The E350 is similar.
Here is a grease seal which costs maybe $6 at an Autozone.