Driveline vibration?

I recently had a 1992 chrysler 5th ave drop into my lap. When its cold, there is a vibration only when coasting, most noticably between 30 & 40 mph. Picking up the throttle even just slightly makes it go away, and its barely noticable after driving a few minutes. Could this be slop in the axles or something else? I can gladly take 5 minutes of shakin to keep this car under a weeks pay, but I hate towtrucks, any ideas?

Probably needs a tune-up. This sounds like engine studder, not drivetrain issues. Drivetrain shakes don’t go away. IMHO, I’d tune it up, and change all the fluids, including transmission. Even if the fluids look good, they could be compromised by mileage or age.