Your plumbings riht in the pic.
\Crankcase line goes to the adaptor with supplies a pulse to the carb.
Tank line should go to black nipple.
If theres another nipple,block it off.
Vent from tank goes to where ever its not in the way.
In a plane i make one coll on top of tank,starp it down.Then i run the line out the bottom of cowl towards ground.
The only time a carb will drip fuel on its own is whe the needle and seat are stuck open or wrn out.
There under the cover with the 4 screws in it.
The pump is on the other side with 1 screw.
Get a different carb or reset the needle.Or take it to a repair shop.Or buy a new one.[from here on RCU or ebay].