While I know nothing of your car's model characteristics, in general the things you describe sound more like an issue with a sticking brake caliper. They can stick (probably on the right side in your case) causing the car to pull to one side. Then the heat buildup causes the disc rotors to warp - that then causes that side to not only pull from the constant friction but then shake.
Get the rotor checked on that side, and if it is warped, replace it and either repair or replace the caliper. I think they can check the rotor warp on the car with a dial gauge.
Note: I am only a amateur mechanic, and my disclaimer is that I know nothing.