Can you turn off both service switches long enough to see what happens with the voltage ?
I would not rule out "leakage" thru the bolted switch, but I would be more suspicious of some type of back feed from the other service.
It seems possible if someone has accidentally fed the same load from both services you could have 120v on the one leg and the other voltages are from backfeeds thru other mutli=phase equipment.
To me, the flickering lights support the back feed situation.
(Would not be the first time many of us have seen one load getting power from two panels.)