The shaking wheel part suggest that there might be an engine misfire, the engine light on suggests that the computer is trying to tell you what's wrong.
Connect the engine to a diagnostic computer to see what's going on. The computer can often give you hints to the real problem.
you steering wheel starts shaking because you probably have warped roders your shaking will get worse at higher speeds and when pressing the brake
A bent, loose or unbalanced wheel. And, if there is no shaking in the steering wheel, then it is most likely from one, or both, of the rear wheels. Irregularity or bulge in a tyre also causes car shaking. I've has this problem once, and it was because the engine mounts needed to be replaced.
What is the answer
There is no light on the steering wheel.
1. out of balance tire 2. Steering out of alignment 3. bad wheel bearings
It sounds like the inner steering shaft. I had the same problem on a 1997 Dodge Ram.
Tires need to be balanced and re-aligned
Tire out of balance, bent wheel, defective tire, or worn steering parts.
At highway speed, tire out of balance or bent wheel. When stopping, warped rotor.
If your steering wheel is shaking you probabley messed you wheels or axel up
A shaking steering wheel means something is wrong. Can be a tire out of balance, bent wheel, or suspension problem. Whatever it is, the shaking it putting stress on other parts and can cause other things to wear out. It is especially hard on wheel bearings. have this diagnosed and repaired asap.
That is typically a wheel balance problem.