Did you use a real Kohler fuel filter and the correct one for your engine?

Kohlers are very picky regarding fuel filters.

Here's a link to real Kohler fuel filters and when you click on each one it will list part numbers and revisions... click here

Normally if if the fuel supply is clean and it had good flow it comes down to carb service like most. If you didn't when you replaced the f filter "run the line" and take a fuel sample to look for debris/ water- then do that now. A touch of water can be drained and a shot of Seafoam might be all it would need. Quick and cheap. My personal experience on CV twins are that they aren't finicky on filters. Kohler spec them varies from 10 to 51 micron. Part of which way to go depends on what is really happening verses what is perceived as a no load surge which is usually still somewhat found under load. If you have a mechanical aptitude with good attention to detail, you may do fine removing the carb and giving it a quick clean. If not, have it serviced.

I replaced the fuel filter with a Kohler filter of the same P/N as the old one. I'll try cleaning the carb and see what happens. Thanks both of you for your time and advice.

might also try spraying some rattlecan wd-40 around intake manifold connections to ck for air leak with engine running. If engine temporarily smooths out, then look for loose components/hardware/damaged gaskets..

It's the Kohlers that rely on gravity feed (i.e. those without fuel pumps) that are so finicky about fuel filters. In any case, always use a factory-recommended filter no matter what engine you have.