Climate control system
There are a lot of tests that can be made--a whole book full of them! I know very little about the CC system, but I do have the Model 201 Tempmatic CC Service Manual from Mercedes.
Sounds like the hot water is continually flowing into the CC heating element. I would first check the heater valve, which is a vacuum operated valve located below the air intake grill at the front of the windshield--to the left of the Windshield Wiper motor as you face the front of the car. This valve controls the flow of hot coolant into the heater core. According to the manual, the valve is CLOSED when vacuum is applied, and OPENED when vacuum is absent. Could be a leaky hose, bad valve or bad vacuum switch, but at least you can start at the valve. If you have a vacuum gauge, you can connect it to the hose to see if any vacuum is present when the heater is supposed to be off.
The switchover valves are located below the instrument panel on the passenger's side of the car, it appears from the picture.
Good luck, it is a complicated system. I would at least make this test before you replace the control unit. You should also check the auxilliary coolant pump--on my car it is on the left side of the engine compartment across from the exhaust headers. It is supposed to pump coolant from the heater core when the heater is on. Check to see that it is off when heat is supposed to be off. If it pumps all the time, the control unit could be bad.
Try to find a copy of the Tempmatic Climate Control Service Manual if you need to really get into the system.