Q: Car won't go in reverse - 2006 Chrysler Pacifica

My car had the check engine for about 8 months now, but has been operating just fine. 3 months ago I got the "service engine" notification on the dashboard and I serviced it twice but the indicator never changed. A week ago I started hearing loud noises coming from the back, and for the past three days the noise became louder, but not continuous. The noise sounds like a motorcycle. I barely was able to drive home last night. This morning I was going to take it to the repair shop but it refused to back out of the parking spot. It goes perfectly fine forward without me having to hit the gas, but when I hit the gas on reverse to go the smell (new as of this morning) and the noise goes off. I looked at the exhaust and system it is 30% rusty, no holes on the exhaust. What is wrong with the car? What caused this? What should I expect when I get it to the shop? How much should I expect to pay?

My car has 184000 miles.
My car has an automatic transmission.