I'm sorry, but I have little familiarity with the 123 automatic climate control systems. The 124 pushbutton unit can often be repaired by simply reheating the solder joints ON THE CIRCUIT BOARD.
My chmiming in about soldering was from my old electronic bench days, simply as a warning in case you are not familiar with circuit board soldering. A 25W or 40W will work okay. I would not recommend soldering anything that is not meant to be soldered.
Sorry I can't help any further.