Its probably just a worn float valve, i would get get a rebuild kit and soak the carb overnight in carb dip, try your local auto parts place for it. Some places carry the container of it thats large enough you can just toss the entire carb into it and put the cover on, but make sure you remove all plastic and rubber pieces from it as well as all your removable parts, jets and what not, clean those pieces seperately and use some small strands of wire to clean out all the holes. I bought a kit for carb cleaning at a local auto parts store years and years ago, but i hear all the time guys mention using piano wire or guitar strings to do this job as well. Along with some small pipe cleaners to clean the jets.
Always shut the fuel off, even if you think you can trust the float valve, its just a two second cheap insurance!
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1984 Honda ATC200ES "Big Red"
1982 ATC200E "Hondie"
1988 TRX300FW "Project Quad" Still in progress....