Can you give us a few more bits of info?

Type of Carb
Has Carb been rebuilt recently

That would help.

Do you know where it's dripping from ?  A quick way to find out is with a match.

"I started looking around and noticed fuel dripping off the ventures."

Do you mean the venturi, in the carb?

If there is fuel dripping, or running out of either of the primary venturis with it idling, it is idling too high, or, as a long shot, the float level is too high.
The venturis need a vacuum signal to pull the fuel up out of the bowl, and down into the bore of the carb. There is really no other way for that to happen unless the throttle blades are open enough to expose the transfer slots that are located just above them.
Sounds like you are having to raise the idle speed to compensate for what is probably a too high float level(flooding). That's why the idle rpm gets higher after driving, as it is clearing out all of the raw fuel. Once it starts idling again, it loads up, and the idle goes down, and you have to raise it again to keep the engine running.
Looks like a carb build is in your future.