And welcome to the 2006 Ram Hemi Club of which I, too, am a member......ha ha. As you can see from my signature I also have a 2006 Ram 1500 Hemi. Mine has 178K miles now and I have not experienced the issue you are now experiencing. Sounds like a broken or faulty seatbelt switch or sensor in the receiver(?). Maybe try getting one from a boneyard (?). Otherwise I suggest checking with your local dealer. I've not read or heard of this issue.
The simplest (cheapest) solution would be to disable the chime altogether. Instructions to do this are in your owner's manual. This won't get rid of the red seatbelt light but will stop the annoying chime.