How did you check the carbs?
What you describe is almost always a result of clogged carbs.
Enough gas is flowing to keep the motor running when it's not under load but as soon as you put it in gear and ask it to work it dies of thirst.
Pull the carbs, take them apart, soak them with carb cleaner, blow out all the (holes) with compressed air, inspect any needle valves and jets (replace if deformed) then install with new gaskets/seals and o-rings as required....
I will assume that you have confirmed that the compression is good and you have strong spark at all cylinders...