If you are getting power, but not the right kind/qulaity of power, my first thought would be that the generator RPMs are off. The genset engne must mainatan the proper RPMs to enable the generator portion to produce the correct alternating current frequency, i.e. 60 cycles/second. If the frequency is off, the voltage will appear wrong to most meters and anything that uses the power frequency to control a function (a tv is one such and clocks are often another) will not perform properly. And in fact the voltage may be wrong, when measured as an RMS value over time.
The other possibility would seem to be low peak voltage from the genset output, but I can't think of any way for that to happen unless a winding is shorted out in just the right place, reducing the output slightly but not enough to be totally out. Seems unlikely, but is conceivable, I guess. Moisture accumulating in the windings could do that, I suppose. You might try letting it run (preferably under load, if you can find something that will tolerate it) for an hour or two to see if it dries out. Light bulbs and heating elements are not frequency sensitive and tolerate low voltages too, but motors and most electronics are not very tolerant and could be damaged. I wold not run any of them until the problem gets resolved.
Sounds like a voltage regulator or slip ring/rotor problem. The voltage output of an alternator is directly related to the excitation voltage to the rotor through the slip rings. If the regulator is not putting out enough excitation voltage, the output voltage will be low too. Another possible cause is dirty slip rings, worn slip ring brushes, or open/shorted rotor windings. The output frequency is a function of the rotational speed and if the engine seems to be running normally, that shouldn't be a problem. If it's dirty slip rings, running it with a heavy load may clear up your problem, but don't use the a/c for the load as you could seriously damage the motor if operated at too low a voltage. Use a couple of those small cube heaters for the load.