Shake of death

Take it to a tire and alignment shop for an inspection. They will look at the ball joints, tie rod ends, track bar, and control arm bushings. There will also be a steering stabilizer that is the likely suspect. It looks like a regular shock absorber but it pulls apart and pushes together equally hard either way. One end attaches to the cross member and one end to the steering linkage. It's purpose is especially important on Jeeps because they call for a REALLY high caster setting, (11 degrees compared to around 3 degrees for most cars and trucks). Caster is what forces the steering wheel to self-return to center after you turn a corner. When caster is real high, the steering wheel comes back to center so fast, it overshoots and goes the other way, then it comes back again and goes too far again. Road forces from moving down the road keep that oscillation going until you slow down. The steering damper stops that. Typically the damping oil leaks out where the shaft comes out, then in effect, it isn't even there. Even if there's no sign of leakage, that damper is the most common suspect.