High speed vibration

I just bought this 89 GT. Nice car but has a few bugs. one that Im dealing with now is a vibration to the rear of the driver. I ve owned several Mustangs but never ran into this before. At 65 mph I develop a vibration which appeared to be connected to the driveline. Put it on the lift and shook the driveshaft thinking the universals are bad. Not so. Universals are tight, but There is a little shaking where the driveshaft goes into the rear of the trans. There is also a little turning (think the mech called it lashing?) of the shaft at the rear end. I consulted a driveline guy who suggested balancing the shaft.$125. If its not that it could be the bushing at the rear of the trans. $125. If not that some type of thingy in the rear end. $200- $500. What do you guys think? anyone run across this before?
 
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Check the rear part of the driveshaft and look for a yellow paint mark. That mark is supposed to line up with the same color mark on the rear axle driveshaft flange. That tells you that the driveshaft is properly indexed for minimum vibration. Also make sure that the dog bone (heavy cast iron weight on the bottom of the differential housing) is present and the rubber mounts are in good shape. The dog bone is a vibration dampener that is supposed to filter out some type of driveline vibrations.
 
1st of all, thanks for the response and I will chk for the weight as soon as I can get the car on my lift..... but is the drive shaft supposed to have play in it when you turn it back and forth. It has about a 1/2 to a 1" play. If it isnt supposed to have play would that cause it to vibrate, and whats the fix?
 
Does it have 3.55's or higher? If so, install an FRPP aluminum driveshaft with the dots lined up as pointed out above...problem solved. What other suspension or engine/trans mount modifications does it have? Could be a pinion angle problem too.
 
Thanks for your imput. I have no idea what gears it has, however, it turns2000rpm s at 65 mph. It is a 5 speed. Is there some way to determine the gears without removing the diff cover. Oh, by the way, I just ordered an aluminum drive shaft from LMR, along with a safety loop and new bolts. Hope that does it! If not I ll look else where I guess. Any other suggestions? The driveshaft remains suspect.