Heres a thread on another site about repairing the connection to the LCD. This unit is absolutely identical to my 043H module right down to the colors of the bulb sockets on the back, and the little green circle sticker on the top.

Audi A4 B5 Climate Control Solder Joint Repairl

I may take one of my 043B units apart and see if I can install its display into the older 043H unit. The front end of the climate control module should be the same, but thats a project for the future.

 

Great resource. I got into the controller tonight, but I did not see any signs of bad solder connections. But some must be loose since all I have to do is push in on the screen and it lights up. I'll take it out again tomorrow and do the multimeter tests. Thanks.