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PetrosA:
The low line voltage for the most part is a power company problem. When looking at changes in line voltage it is important to know the corresponding current changes. Your additional information indicates the source is a single phase center tapped transformer. Let's assume the total loop impedance at the main panel and looking toward the power company is 0.1 ohms for neutral, the transformer, and one hot line, then if you placed a 10 A load on this circuit at the main panel the voltage should drop 1 V, and 100 A would cause a drop of 10 V. So a 1500 W 120 V heater can be a useful test tool. It is about 12 A.
Let's review your ground voltage problems.
With the main panel disconnect switch off you measure near 0 V from the transformer ground rod (presumably the same point as the transformer center tap) to any point in the soil or water pipes around the home. My interpretation of this is that there are no stray power company ground currents of any large magnitude.
When the main disconnect is closed, then large voltages are measured in the earth, on water pipes, and on drain pipes. This means if there is no large voltage drop on the neutral to the main panel from the transformer, then the neutral at the main is not bonded to the water system. Therefore, I believe the water piping system, etc. is floating and somewhere there is voltage leaking from one 120 V hot line to the earth and/or the water piping system. If there is an electric hot water heater, then check it.
If I have current in the earth flowing in a north south direction, then two probes aligned in a north south direction should produce a maximum voltage. In an east west direction voltage should be a minimum.
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