Continuity check....

When I discovered a similar problem at my wife's old house, it turned out that the weather head had the neutral wire corroded out. This caused a floating neutral that would not feed 120 volts to either leg unless the load on both legs was about balanced. When checking the voltage on one leg, turn on something that is plugged into that leg and see if the voltage drops. If so, then plug a similar load into a circuit on the other leg and then see what effect it has on the voltage again. If the voltage keeps changing up or down, then you likely have lost the neutral leg from the pole to your main box, likely near the weather head.