1993 Chevrolet Cavalier it shakes badly in parking mode

Hi,
I have 1993 model Chevrolet Cavalier RS 2.2Lt Auto trans.
My problem is when I start the car I've noticed annoying vibration which I can feel through the gas pedal and steering wheel as well. Vibration goes when I gain speed but I can stiil feel it on accelerator pedal. Could it be engine mounts collapsing?
My second problem is about stalling. When the engine is hot it doesn't downshift when coming to a quick stop. And then it shakes back and forth eventually stalls. After waiting for a while I start it again but when I put the gear into driving mode it stalls again.I can't move it unless I accelerate right away after put the gear into drive mode.I was told it might be TCC solenoid or Torque Convertor problem?
Please help me.
Thanks