I think the wobbling you're experiencing despite having the wheels balanced is one indication that the ball joints or bushings or both are in need of replacement. Similarly, vibration and wobbling when braking from high speed (not a panic stop, just braking such as coming to a toll booth or something like that). A rule of thumb that I've found to be fairly accurate is that the ball joints last about 50,000 miles, and that's with easy to moderate driving.
It'd be worthwhile to take it to a reputable mechanic or alignment shop to get a good look at the undercarriage, at 200K there are probably a lot of tired, cracked, worn bushings and whatnot. You should also check your engine and transmission mounts, if they're worn they can also send vibrations throughout the car that might be interpretted as wheel related, but aren't.