Partner,
There is no simple answer to your question. I could tell you F1 and F2. And the next person could tell you something completely different. What you need to do, is check every wire. Its a PITA, but if you do not systematically go about checking, you are wasting your time. In post #4, notice the word, "somewhat of the same problem" Every problem can LOOK the same, and have DIFFERENT symptoms.
In post #3, the subject was brought up of frequency. Also important. And you never answered the question. Is the frequency staying at 60 hertz when you pump up the voltage? As the load goes up, are you also pumping up the frequency?
At no load, how can you get 124 volts? I would turn down the voltage. I am using the generator in the 120 V mode. At 24 amps, I get 110 V. At 15 amps, I get 118 V, and at no load, I get 124 V. This doesn't really tell us the right info, in the right sequence. Normally you start in the other direction. You start the set, bring it to 120 volt, 60 hertz. Then apply load. What are the readings? And so on.
You got to help us, help you.