I have a 2001 Audi TT Quattro that wont shift into gears

What is your question?

I noticed lately that my car won't shift into gears when the car is running, but once the car cools off it shifts into all the gears. I noticed the problem initially when I was trying to reverse and there was a smell like plastic was burning. I've invested a lot of money and have taken this car to several mechanics and they cannot figure out why the gears get stuck. I'm going broke trying to resolve this problem. In the last 10 months I put a new engine, put in a timing belt, new battery, clutch and had air conditioning fixed. The timing belt broke, I store it during the winter months and bring it out in the summer months since this initial break I've had nothing but problems.

What makes this problem better or worse?

It's worse when the engine is hot

How long has your 2001 Audi TT had this problem?

A couple of weeks