I have a Honda weed trimmer/brushcutter model UMK422 that will start right up on the first pull but then quits running after 15-20 seconds, then won't start again until it has sat a good while like half an hour or so. Then will start right up again but will again always quit running after 15-20 seconds. Has good fresh gas and a the gas filter is fine, and the gas lines are clear and in fine condition. Assuming a carb problem I have at least disassembled and inspected (it's a Walbro carb) and the valve needle was fine and the pump diaphragm isn't obviously bad looking (no rips and not stiff anyway). The gasket(s) look fine too. Does the symptom I describe sound like it could be remedied by a carb rebuild maybe, or new carb? What's got me rather baffled is how it'll start up fine and run, but then just quit every time as I described. Thanks for any comments.

Tecumseh 10 hp engine HMSK100 - 159208T on a 20-year-old MTD snowblower. The engine would only run at full choke, full throttle. The thing was a gift, but I decided to rebuild it and try it this winter. I installed new carb, spark plug, fuel lines, cleaned and lubricated, etc. Starts with half a pull... but still stalls out when I open the choke even halfway. My guess would be the idle air adjustment, but I welcome any guidance. Thanks. BTW, why is there no air cleaner - or even any provision for an air cleaner - on this engine???