I have an Audi A4 and have never had an issue with the towing electrics. When I hooked up this year the right hand indicator stays on solidly when the Brake pedal is pressed.
I have put a multimeter on the sockets and am expecting a dodgy earth. Attached is a pic of the results from the multimeter. Are there any sparkies who would be able to look to the results and let me know if I can simply run a new earth lead to the socket???
Take the socket off the motor and check the connections at the back of it. Usually the problem is water got in causing corrosion and shorting across the terminals causing this.