If it runs fine once it's started and will sit and idle okay, I don't think the carb is flooding the saw. Does it ever cut out when running and die or is it only a problem when you shut the saw off deliberately? If so my other suspicions would be a bad or incorrect-model spark plug (if you've put a new one in recently this is the most likely culprit), or a plugged fuel tank vent.
And FWIW it is possible to flood a saw with some primers because they shoot fuel straight into the carb throat through an auxiliary jet in the carb. In such a case though the saw will usually fire for a couple seconds until it pulls too much fuel into the crankcase and floods itself.
Are you using the same starting procedure (choke, prime, pull) to start the saw even once it's warm? Because once you've had it running and warmed up, you shouldn't need to use the choke OR primer to re-start it within a half hour to an hour of it being run, and probably just the choke but not the primer for quite a while after that even.