chaimv,
I think the only sensible way to approach one of these old carburetors is a tear down, bath and a rebuild.
There are only a few external adjustments, and they most likely wouldn't address your problem.
The rebuild kits are inexpensive and the skill level required for the rebuild is only moderate.
Keep in mind the carburetor has not been deemed public enemy #1 yet.
With these kinds of problems, the SOP is to do a tune up and replace as many of the 'wear items' as you can.
The car is over 50 years old and probably needs some TLC. Instead of finding a specific item that is at fault, it is more expeditious to tune it up, then see what doesn't work!
If you could analyze the problem in finer detail, the search would be narrowed. Good luck
Stuart