I'd take the bolt/jet that hold the bowl on the bottom of carb off
and let some gas run out till you are sure there is no fuel delivery
problem to that point. Be sure the cap is venting properly. I've been
finding a lot of water in mower gas tanks and carbs (including mine).
Like others said only 100% gasoline in the small engines. Adjust the
valves, it's not that hard to do. You should also be able to keep it
running by taking off the air cleaner and choking it a bit with you
finger to keep it running until it warms up. These things are like
playing amusical instrument. Stay after it.. check the cables and make
sure the safety brake isn't dragging. If you want to try it you might
try finding a jet/carb bolt with larger jetting or very and I mean very
very slightly make the jets a little bit bigger using a welding torch
cleaner although that isn't supposed to be done or needed. Best advice
I know is be sure to use only 100% gasoline. I once had a weird one,
the spark was jumping to ground because of a cracked coil, but it would
fire and almost run. Sometimes for whatever reason a good old spark plug
works better than a brand new one. Don't give up... just one more pull
on the rope and if the rope breaks... well fix that also.