I have an 84 500SEL with about 260k. Three days ago, I noticed that when I first start my car in the morning, the brake pedal is a little spongy. As I drive to work and the car warms up, the brake pedal gets firmer and firmer, but at 50mph, the car starts shaking pretty violently. I have had the tires balanced, and the pads/rotors checked, and the problem still persists. Could this be two separate issues or are they connected somehow?

When the shaking/shimmy/vibration happens, I pull off to the shoulder, put it in park and turn off the ignition. When I restart the car, it seems to be OK for a few miles, but then the problem happens again.

Any ideas? I'm not much of a mechanic, but maybe your advice could point me in the right direction for repairs.