OK, That is odd.
It should not feel that "floaty" with those miles.
Does the ride height appear normal?
Logically, shocks or springs.
Get car in the air, and check both front and rear springs for any splitting or cracking of the spring itself. Typically, the car droops noticeably if there is an issue with the spring.
Next, do the usual bounce test at each corner to see if one or more shocks has too little resistance in either direction.
Check the condition of all of the suspension bushings, front and rear.
Alignment, if all of the other check out, then check the alignment of particularly the rear wheels and the camber settings. If too positive, the car can seem twitchy, which sometimes is described as unstable. Refer to Vette Brakes and Products for their suggested street settings as a good reference.
My first suspect would be the shocks might need replacement, although the mileage seems low for that.
Most often, that causes the sea sickness symptoms.
Let us know what you find.
Marty