I had a similar problem after cleaning the carb on our (now sold) LT-A50. During reassembly I adjusted the float according to the Suzuki service manual, and afterwards it acted very much like you're describing. I opened up the bowl, put the float back to the (contrary to factory spec) earlier position, and the problem vanished.
When you're idling or "lift the back end up" you're not really flowing that much fuel. You hit the max RPM's pretty fast with very little fuel flow because there's almost no load on the engine. But once you're actually trying to ride it, the fuel flow goes way up - and with the float adjusted I think the bowl was starving out of fuel. "Mis"adjusting it back to where it was kept the bowl full and thus stopped starving the engine for fuel.
Anyway, I'd check the float and reexamine the fuel system and carb circuits for contamination. While you're in there, take off the fuel tank and clean its internal filter, all fuel hoses, and the fuel valve (petcock). None of this is difficult and will give you peace of mind knowing that your fuel system is solid.
Report back!
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