First, I think you need to say adios to that mechanic. To blanketly say "they have a flooding issue" and not address the issue is bs.
Second, and I'm going to be blunt, you made a mistake buying this car for use as a daily driver. It's 60 years old, clearly not in the best condition and is not reliable at this time for a daily driver. It can be with lots of work but not right now. You do not want to have a need to keep it on the road because it makes you rush and take shortcuts fixing things which doesn't always help make it more reliable. It's likely going to have to be off the road for days or weeks at a time in order to fix issues properly.
The bad thing with vehicles this old is it is hard to find someone who actually knows what they are doing and it is best to simply learn for yourself. That in mind, the first thing you need is a shop manual for the car. They can usually be found as reprints of the factory manual and many can now be had on a DVD for use on your computer.
You mentioned a carb swap. Given your experience, I do not recommend that. You will need to change other things in order for it to work properly and you will need to be able to tune the carburetor. Before you do all that, you need to have a pretty good understanding of how things work. For now, the best course is to simply get what you have working well and to learn as a result. Save the carb swap for another topic, another day, when you have a greater understanding.
RE the leaking carb, you need to determine where the carburetor is leaking from, why, and fix it. Probably much more vague than you want, but it's nearly impossible to diagnose something like that over the internet. Since you are inexperienced, I suggest you send the carb to an expert for rebuild. You can also check your shop manual, read about it on the internet and if you feel comfortable, do it yourself.
What you describe on the exhaust sounds like the heat riser. I don't know what liquid could be leaking out of there. Maybe oil from worn exhaust valve guides.
What color is the wall of smoke? Black is fuel. Blue is oil. White is coolant.
Another thing that helps with internet diagnoses is video. Get a video of the problems, post it on youtube and include a link here. That way we can see what is happening and be better able to help.
Lawrenceville, GA