Your generator has a governor system that will keep the RPM's up when a load is applied. On my Onan, there is a very small hole in the intake system that samples the engine vacuum and will speed up the engine when it is under load.
If the generator has not been used very often, it is possible that something has plugged up the governor portion of the engine that does not allow it to compensate for the additional load.
I would check the output voltage and see if it remains constant as a load is applied. If it drops greatly under a normal load, then it may be a generator problem, especially since you said the leveling system worked fine with shore power.