I have a 78 and liked the ZS carb except for the many problems I had with the water choke. You might want to consider dumping the water choke and converting it to a manual one. It's not too difficult to do and will run just over $200 if you order from MOSS. There are cheaper alternatives out there.
Parts from Moss are 366-710 ($97.95) & 386-325 ($118.95)
In regards to rebuilding the ZS, kits run about $50 I think. It's not difficult just get a manual like "Hayes Weber Carburetor Manual" and it will walk you through the process. I figure if I could rebuild one, anyone can as I was a newbie and mechanically stupid but was still able to figure it out. One thing though - I rebuilt mine on our kitchen table over a weekend. You might want to avoid doing this if you have a wife or significant other. All I can tell you the midget is easier to figure out then women, well then again maybe the electrical parts of the midget are about as confusing as women. I am thinking they may be wired by Lucas as well.
Running "Rich" could be a malfunctioing choke as you suspect or an incorrect or worn Needle, etc. Hard to say and may take some investigational work. My water choke kept pouring gas out of it no matter what I did. I went through 3 ZS carbs trying to figure it out but never did. I finally switched over to a SU HIF-6 carb and am having fun working on getting this setup tuned to maxium performance. At one point I just took off the water choke assembly completely. The leak stopped and I was actually able to run the car on a warm day without problems. You might consider removing it to see if it continues to run "rich" and thus rule out the water choke. Just a thought. Others may have better ones.
Anyways enjoy the process. I've learned alot about car's working on the midget and even more about myself. Good luck.