Try changing spark plugs and wires
If a car shakes during acceleration the brake rotors may be damaged. If the rotors are damaged they will be unstable as the tires rotates making the whole car shake.
Time for a serious tune up.
If the shaking is specific to acceleration and the shaking is very rythmic (not random) it is most likely a fault in the ignition system. Usually a tune up or ignition coil replacement will solve this problem.
You need to find out what is killing the O2 sensors. I suspect it is running rich, or is leaking coolant into the combustion chamber. Have the computer scanned for codes. If you are having to add coolant, suspect a blown head gasket. Plugged Cat on the exhaust!
A Dodge Neon may experience shaking during acceleration due to bad spark plugs, incorrect ignition timing, or a failing transmission. Shaking can also occur from worn motor or transmission mounts.
If your car shakes while accelerating check your motor mounts. Then check your transmission.
definant possibility. with a broken motor mount, the engine will lift up during acceleration thus causing vibration.
Engine and/or gearbox mountings failure?
it won't shift gears right. may shake spit and sputter. check engine light may be on or go off and on all the time. rpms may jump up and down while it shakes spits and sputters. bad acceleration. That's what mine did to me and it was my transmission.
My 93 ford ranger with a 4.0 V6, shakes and sputters between 30mph and 60.mph, just finished replacing both head gaskets
Halogen fluid. When it gets low this can happen.
need to clean up the throtle body first.