Intermittent harsh shaking while driving & braking

Having an intermittent shaking issue with my car. No engine codes thrown, and regardless, engine has plenty of power and transmission shifts just fine. The car will run just fine, but sometimes after making a stop, when accelerating AND getting to speed, the car shakes/vibrates profusely. Got on the interstate yesterday doing 75mph and the car ran perfectly smooth. Stopped at a gas station, and as I pulled out of the stop and started going onto the "on" ramp, the car started shaking. The whole time I was getting up to speed of about 65mph, it shook. Drove for about 10 miles, and I thought the car was going to explode. But, slowly, the vibration slowed and eventually stopped, and the car went back to driving perfect (still on the interstate). Same trip, got off the interstate, and as I slowed, the shaking started again. When I go to stop, the vehicle lurches back and forth, and almost seems like there are no pads on the breaks. I'm stumped on this one because of the intermittent aspect. Local mechanic mentioned bad motor mounts. Any thoughts?
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Comments
Was it the motor mounts?
So the best we have right now is a good description of the problem that a technician has to get to occur. Then proceed from there as required.