Engine has ticking sound, oil light came on, and it's full of oil

Normally high frequency ticking is the valve train located under your valve cover. This ticking, if sudden, means oil starvation. The valve train is the furthest component to get oil and is the first to notice low oil pressure.
If your oil level is good then I would look at the oil pump. Please take this serious, the oil light does not offer much info; you need to know what the oil pressure is before you go any further.

You might start to see if you can get an actual oil pressure gauge hooked up, to see what your oil pressure really is. If it is low due to being worn out, or failing oil pump, or waxy build up in the engine, you can destroy an engine, if it isn't already. Any good garage or mechanic with the tools should be able to hook one up to see what pressure your oil pump is actually giving.

The ticking could be a spark plug wire shorted out to ground of which you can watch in the dark while the car is running and see if you see a plug wire jumping to ground which will look like a lightening bolt hitting the ground. It could also be a lifter making noise. Could be a exhaust leak. What is your oil pressure? Does your gauge tell you pounds of pressure or is it just a light?