cub left me afoot

1958 regular cub 12 v conversion? this one is 12v i thought all red cubs were 6v , left me walking yesterday for awhile, now ive been using this to mow with and its been very hot approaching 100 and thats a first, im 6000 ft up in the mountains! now it was in some bumpy terrain, and ive been leaning on this cub a little and having to go to full throttle to mow, should have used a larger tractor but i needed to work in a bunch of trees on a acre, since i cleaned the gas tank out completely last summer this one hasn't missed a beat not 1, has gas in the tank, almost full in fact, had gas when i removed the plug on the bottom of the float bowl as well as in the clear sediment bowl, spark was real good, [ guess how i know] now after a few minutes it started right up and idled, when i started the belly mower and advanced the throttle, it quit again,would not hit a lick, 10 minutes later it started up and idled, so i idled it up a steep hill and back onto my trailer and headed back to the farm, so the question is it is obviously is a fuel problem ,is the thing just getting hot and vapor locking? or is there a high speed passage in the carb that may be blocked with a piece of sand ect? this cub was in the desert and its tank had some sand in it last year i cleaned it real good, but anything is possible what should i look at here?

Take the fuel line off at the carb and see if you are getting a full stream of fuel. If not clean sed.bowl and tank. I doesnt take much fuel to run out of plug at carb.

Not knowing a lot about how your cub is set up it almost sounds like the coil my be getting hot and cutting out. If it is 12 volt conversion does it have a 12 volt coil and or a resistor to cut the 12 volts down while the ractor is running? It could be the condenser also. Just went through that with a Massey Harris I was workinbg on. Put the tractor together with new points and condenser, ran great for about 1/2 hour then would within a minute go from running fine, to missing, to finally stopping. I thought it was a fuel issue as the electrics were all new. Finally replaced the new condensor with another new one and it has not missed a lick in a month.
If you still have a magneto on it, it could be the coil in the mag getting hot and shorting out.

Make sure the vent on the tank's cap is open. They're especially prone to starving out when the tank is full of fuel.

As already mentioned, open the drain plug on the carb the next time it stalls and see if you have a good flow. If so it's most likely ignition.

could be gas, sounds like coil overheating