Since I don't know what you have done so far I'll recommend a couple of things to do and you can go from there. Do you have a strong water stream coming out of the motor when it's running?
First I would replace the impellar (waterpump) they are cheap and should be replaced every couple of years or so. Manufacturers recommend every year but I don't know anyone that does that. If the motor has set for a while without being run the plastic impellars dry out and become brittle.
Second , I've had this same problem you are having with a couple of motors with the choke problem. They have a carb cleaner that you can buy at most marinas , disconnect your fuel inlet line of the carburator and spray a few squirts in....let it sit for a few minutes and then take an air compressor with the blow tip and give it a burst of air in the fuel inlet. Listen for the piston in the carb to seat because they stick from the oil sludge and get gummy. The carb cleaner should loosen it up and the bust of air should reseat it. I've had to do this twice on two different motors and it worked both times. The second time I just did the air and it worked without buying the carb cleaner.
Give it a shot and let me know how it works.
Scanoe