Kohler SV600

My guess is that the float needle valve isn't shutting off the flow of fuel from the fuel tank. This is usually because there's junk stuck in the valve, that is, the needle isn't seating correctly.
 
Remove the carburetor, drain it, flip it over and remove the fuel bowl. You should now see the float (looks like a life preserver), and when you look closely, you'll see that it controls a stem (the needle) close to the hinge point. Remove the bar which forms the hinge, lift the float, remove the needle, and clean it, then clean the hole into which the needle fits. In all likelihood, you'll have found garbage and sediment in the bottom of the fuel bowl. Clean it. Reinsert the needle, reinstall the float and hinge bar, but don't close the carburetor up yet.
 
You'll see that the float, as the bowl fills, rises and pushes the needle into the valve. To test that it works, flip the carburetor upside down, which would close the valve, and blow into the place where the fuel tank line connects. You should be able to blow when the carburetor is right-side up, and not be able to blow when it's upside down. You don't have to blow so hard it makes you dizzy; just a little pressure is enough for a test. If it passes the test, close it up and reinstall the carburetor; if it fails, check the needle valve again.
 
You don't need to replace the carburetor.

For the Nikki carb, the little o-ring seen separately (SEALING BETWEEN THE CARB BODY AND PLASTIC PIECE) IS usually the cause of the leak.
 
IME, place it on the plastic piece, then put just a touch of di-electric grease on the outer edge of it
so it DOES INDEED slip into the aluminum. A missing, rolled, damaged o-ring will leak..  
 
 
With that said, on my 18 HP Briggs Zero turn, I replaced the Nikki with this Walbro:
https://www.jackssmallengines.com/jacks-parts-lookup/manufacturer/briggs-stratton/briggs-stratton-engine/200000-399999-series/31h700-to-31h799/31h777-0202-e1/carburetor-kit-carburetor-overhaul 
 
Check the part #, pic's etc and see if it'll fit yours...