Most likely the seat in the carb needs to be polished.
What I did on my 350 projects is add a cheap inline shutoff valve to the line between the tank and carb. Replacement petcocks are pricey.
To the carb, for some reason the 350 Rancher carbs are horrible about leaking. If your float is good, take a qtip and put some polishing compound on it, or even toothpaste, chuck it in a drill, and on low speed polish the seat in the carb. I also usually scuff the rubber tip on a needle valve with some emery cloth or steel wool.
If that doesn't work, get a tiny polishing stone from a dremel kit, with a conical end. Grind the stone down on the sides where it will fit up in the seat, and USING YOUR HAND, turn it back and forth a few times to really polish the seat well, where there's a perfectly smooth, clean surface for that needle to seat.
If that doesn't work, buy a new carb. Don't get aftermarket Chinese crap. Get an OEM carb, preferably new, because if you get used it will probably have the same issues as the one you have.
2005 Foreman Rubicon Camo
Perfex, 27" Mudbugs
Heated grips and Thumb , HID's
5K Viper Max winch
2009 Foreman 500FM Green
Perfex with HL springs
27" Execs on Rubicon rims
Heated Grips and Thumb, HID's
4K Viper Max winch
2006 Foreman 500FM Green
28" Mugbugs, stretched, Perfex, HLsprings
3K Superwinch winch
2009 475 Rancher AT Tan
26" Mudbugs on Rincon rims
Warn winch. Foreman pod light with HID, SRA gear reduction