Lift / Slapper bars on my Early Dart Dragcar

This article here might help to shed some light on your situation.

If it were me, I'd invest in S/S springs at a minimum. They have the extra leaf that in theory is supposed to counteract the torsional rotation form the driveline torquing over and unloading the driver's side tire which may be causing your sideways motion. If you have worn out springs this will be magnified when the tires hit. Do you have a torque strap on the engine? Frame connectors? All bushings and mounts good?

FYI, ESPO makes decent H.D. springs for like $200, U.S. made too. You can ask to get them arched 2" over stock for clearance issues. I had them on my old '65 Valiant, worked good.