Nope. My '06 DTS has none of the symtoms you are describing. When mine is running, you feel nothing thru the steering wheel. There is no excess exhaust noise and no moaning and groaning on acceleration. It does have a tiny bit more exhaust noise than other Cadillacs I have had.
There "IS" a nice sound at full throttle, after 4000 RPM all the way to 6500 RPM. (at least to me)

(I call it the Northstar Song)
I would take it back to the dealer. QUICKLY. There is no reason it should be moaning and groaning unless something is wrong with it.
Jim from Texas