You should be able to close the needles and run it dry, then gradually open them until it runs right without flooding, though if it has a bypass jet, then that may not help. I had a wood splitter do similar things. You had to be careful not to use too much choke or too many pulls with full choke to the point of: pulling with no choke, then pulling the cord with gradually more choke. Otherwise you'd risk flooding it or getting hard kick every time you pull the rope if it wasn't already flooded from shutdown the last time it was used. I started lowering the throttle slowly, let it idle, then shut it off to help prevent it from being flooded for the next start. Not long after it needed a spark plug. I don't know how the wood splitter story helps, but it did seem to have a similar problem, though totally different engine.
I wonder if there are any compatible cheapo chinese carbs you could try?
Yes, but is it set correct in height?
If its too high it may leak when you start it. It can still hold tight when testing but flood as its open too much.
What needle setting is it on now when it floods?
Is intake tight?
On side won't do it. Flip it over and pull without plug and kill switch activated.
After that, close fuel and run it dry.
You likely have fuel in muffler too so remove it and burn it dry.