Heavy car wobble & engine light popped on today..?

A wobble could be caused by many things independent of the engine light:

• Unbalanced, loose or flat tire
• A suspension problem like a bad shock, worn bushing, or cracked component
• Loose engine or transmission mounts

That is just some of the most common causes. The engine light being on could be significant as it points to the engine not firing on all cylinders. This is known as a misfire, which can be caused by:

• A lack of spark in a cylinder due to a bad spark plug, loose or failing spark plug cable, ignition coil problem or other issue in the ignition system. This would likely be a problem with a single cylinder
• A bad or clogged fuel injector in one or more cylinders
• Clogged air filter
• Vacuum leak
• Fuel pressure problem
• Fuel contamination
• Stuck engine valve
• Sensor failure

The car has a computer which will probably tell you what the issue is using On-Board Diagnostics (OBD). These codes can be read by a reader, most auto parts stores will do this for free. But first I would open up the hood and look at the engine myself to see if there's anything obvious, like a loose spark plug cable, or one that's soaked in oil.