I have an 2012 Audi A3 Sportback 

Just under a week ago I got stuck in a flood, the water was up to just below the door. I tried turning it on etc whilst in the flood (stupid I know) but it didn't turn on. Once I was pulled out, after many attempts I got it to start and drove it home. There was a lot of light grey smoke coming out the exhaust the whole way.

It still turns on now, however it is very very shaky and there is still some smoke coming out of the exhaust. The engine is heavily shaking. The air filter was damp so I removed it to dry.

Do I need a new engine, or is it possible to dry the engine out? 

Virginia.

A wet air filter suggests water may have been sucked into the engine, so it is very possible it has been damaged and will need to be replaced.

It is equally possible parts of the electrical system have been soaked and no longer work as they should. Water and electricity do not mix.

Your first course of action should be to get the car diagnosed on a decent tester to get an idea of what is wrong, and hence how much it will cost to put right. A new engine would cost a few thousand pounds and would probably be unjustified on a 12 year old car. A secondhand engine would be more viable if you are keen to keep the car. Wet electronics may be cheap or expensive depending on precisely what has to be replaced.

Whatever the problems are, get a diagnosis before deciding what to do.