Carter 2BBL carb- I am stuck

The fuel may just be evaporating out of the bowl. Carter's are notorious for this, and Edelbrock's aren't much better. The bowls are vented, and if they sit long enough the fuel can evaporate out. It still happens on my Challenger (carter 650 4 barrel) if I let it sit for more than a couple days. Overnight is pretty fast for it to be evaporating, but I don't live in AZ, and I know it can be pretty stupid hot there, even overnight. So, I wouldn't be surprised if it was just a bowl vent causing that issue. I would check to make sure it wasn't draining into the intake and washing down the cylinders as was already suggested, but it wouldn't surprise me if it was just evaporating.

As far as the idle mixture screws not doing anything, it IS a vacuum leak. It may be internal to the carb, but it's there. There are a lot of places a carb can have a vacuum leak internally. One of the more interesting spots is at the throttle shaft, they show up there on old carbs because the shafts are harder than the carb metal, and eventually wear out the carb. Which leads to a vacuum leak that's pretty hard to find, and not easy to fix on most carbs. There's also the rebuild kit gaskets. They don't always match the originals, and even a small difference can cause an internal leak that will be very difficult to find. When you rebuilt the carb did you compare the new gaskets to the old ones?

Personally, I'm not a fan of the Carter 2 barrels. I've tuned a lot of carbs, and I just don't like them. Carter 4 barrels are no problem, but the stock 2 barrels have never worked all that well for me.