Your best bet is to find someone with VCDS Diagnostics (I have just ordered mine for my A6 but not cheap, goto rosstech), this will tell you if anything electronic / electrical has failed.
The A4’s heater system is notorious for failing / sticking, this is mainly due to excessive moisture being circulated / dragged in. it’s an easy job to strip out and check the controller, but a much bigger job if the matrix behind the dash is the fault.
If it’s a conmtroller, you can pick them up cheap enough off ebay.
The best Audi / VAG forum i’ve used is here: –
Audi Forum
Choose your model and ask some questions.
Request here for someone who may be close with VCDS, most people on there withh check you car over for the price of a couple of beers.
VCDS