Typically what I would do with this sort of complaint that's brought into the shop is to remove the carburetor and rebuild it. You should soak it for an hour or so in a commercial carburetor cleaner, not the spray kind. It comes in several different sizes, most common a 1 gallon can and is available at auto parts stores. Before you do that though, loosen the gas cap and see if that helps. Generally a bad cap will only allow the engine to run for a minute or so and then shut down, but I always check the cap first.
You should also purcase a carb kit and replace the gaskets, O-rings, needle and seat or other parts specific to your carb. You will need the model, type and code numbers off the engine to purchase these things. Those numbers are probably stamped into the flywheel shroud or on a plate on the shroud or side of the engine. The engine family numbers, the numbers you sent, won't let you look up parts. If you can give me those numbers I can give you the part numbers you need.
I would replace the fuel line and any fuel filter and rinse out the tank as well. The alcohol blended fuel we have to contend with now causes the rubbers found in many small engines to degrade at a fast rate. Many times it's tiny pieces of fuel line clogging up the carburetor.
There are several things inside the carb that can be causing this, a clogged air circuit, clogged fuel nozzle, incorrect float level, insufficient fuel reaching the carb. But being that this is a generator and is something that has to work when needed, your safest bet is to eliminate all causes. I'd also replace the spark plug and change the oil and air filter too if you haven't already.
Another thing that I've come to use a lot in the last couple of months is SeaFoam. I've been adding this to the gas on a number of my personal engines and some commercial equipment we maintain, and it's a product that really does as advertised. Just follow the label directions.
You're right to run out the gas when you store the generator. That's the preferred way nowadays.
Thanks and let me know if you have any questions, PK.