Sounds like either you have a bent rim(s) or your tie rods are going.
check the motor mounts
Loose Motor Mounts
Wheel balance, Worn steering parts, Worn suspension parts, Tire damage,
For serves, volleys, slices, overheads, or certain ways of hitting ground strokes
Check or replace axles, worn axles are primary source of shaking when accelerating.
you probably have a bad ball joint.
My Dodge Charger doesn't shake...
A 2002 Dodge Caravan may shake back and forth while driving due to several potential issues. Common causes include unbalanced or damaged tires, worn-out suspension components, or misaligned wheels. Additionally, problems with the drive shaft or issues with the brakes, such as warped rotors, can also lead to vibrations. It's important to have the vehicle inspected by a qualified mechanic to identify and address the specific cause.
You need a steering stabalizer and if you have one it may need to be replaced.
Probably the throttle position sensor.
Warped brake rotors........
rear brake drums or rotors are out of round.