Auto is in drive but still has forward motion with engine off

Have you checked transmission fluid with vehicle at normal operating temperature and on level ground?Also is the check engine light on.If so hook up OBD 2 auto scanner to pull error codes.Might want to check with dealership to see if vehicle has cable as part of shifter limnkage.

do not muck about get the car to an automatic transmission specialist. Sounds like serious malfunction of your auto transmission. Could be something to do with either a transmission cooling system failure or damaged bands in the auto transmission .
Check the auto transmission fluid level on auto transmission fluid level dip stick may be low ?.

Is your radiator fan working the way it should? If car cools down with forward motion it seems that the forward movement is doing the cooling and when you stop there is no cooling.

Using an automotive dolly can be risky, If the vehicle is front wheel drive then dolly the front, if it is rear wheel drive then dolly the rear. If it is 4 wheel drive and late model with no neutral shift position dolly the front and remove the rear drive shaft at the rear axle. If the vehicle is older 4 wheel drive with neutral selection for the transfer case then simply shift to neutral and dolly the front axle. Most transmissions whither automatic or manual will be damaged if the engine is not running while the vehicle is in motion.
the engines crankshaft or the transmissions input shaft must turn to lubricate the transmission unless the transmission is automatic and has a rear pump that can lubricate the transmission if the transmissions output shaft is turned in normal rotation for forward direction of travel. This is the only way to to assure there is no damage to the drive train. Be sure to follow all recommended tie down procedures. remember the dolly rotates under the vehicle so one or the other of the wheels must be able to move forward and back to allow for corners. Try to always choose a path that allows forward direction unless you are an expert driver with experience and able to back a farm wagon type axle trailer (like covered wagon) going backward will be a challenge even for someone experienced pulling a bumper pull trailer.
To get the vehicle onto a dolly have some friends help push or use a cable puller to move the vehicle onto the dolly.

Sounds like a fairly major transmission problem - go to an Auto Transmission Specialist - don't waste your time with an ordinary mechanic.

get the OBD codes read as will point you in the direction of the fault

The pump seals in the transmission are worn and need to be replaced(rebuild or replace transmission)

Your transmission is worn out and needs to be rebuilt.

If your car is only a couple years old,& you never said
what you have,then maybe a test drive or diagnostic
approach at a local trans shop, will be what you need

Are the engine rpms climbing when its jerking forward? seems like it could be a Transmission torque converter problem.

wow looks like your gear box just blew up
is it an auto? my guess it is an auto.
one of the cars herd did the same thing
with this car tho the over drive stopped working (4gear) so he was luckily and dident have to pay a sent as he never went over 100kph
this may be the same with you
if it is an auto put in into 1 or low then try to take off if this doesnt work you may need a new auto gearbox