Hi,you may have answered your own question,the first check is to test your engine mounts,all three, trans mount too,it is helpful to
have a friend in the driver`s seat,applying the brakes and set the parking brake, then start it up and put the trans in gear and push down on the gas pedal,first in drive then reverse and see if the engine torques up off the mount,the rubber may be deteriorated and is separated from the steel,be careful and don`t put yourself in front of the car just in case the brakes won`t hold the car. good luck.