Buick Repair Questions? Ask a Mechanic for Answers ASAP
If it were the lower control arm bushings the front end would also be loose and the steering wheel would also most likely have play in it.
If it only happens during the first few minutes of driving then it may be flat spots on the tires from sitting all nite. I would test that theory. If the seats and floorboards shake then the problem is most likely with the rear tires/wheels. If the steering wheel shakes then the front.
Depending on which, jack up that end and place jack stands under the vehicle for all nite so the tires don't set on the ground all nite, then the next day lower the vehicle back down and immediately take it out for a test drive.
Also, if a shake starts out almost immediately, say like 5mph and just gets worse and worse the faster you go then it is most likely a bent rim/wheel. if the vibration/shake doesn't occur untill about 40mph and goes away over about 50 mph, then that is a tire/wheel or tires/wheels out of balance.
If any play in the steering or wander then also make sure to check front end parts for being loose/worn and also the rear for being loose/worn. If it pulls to a side then that is either a front end alignment problem or the racking off and the rear end crooked because it's loose or something bent.
If it makes a noise going around tight turns forward/backwards at slow speeds than that is most likely worn cv-joints. A violent shake could be a bent or unbalanced driveshaft if equipped with one.
Tread separations in tires could also cause some problems that may appear to be balance problems.