Does a higher gauge electrical wire impact voltage level?

Wire gauge will not affect no-load voltage.  Your reported voltage swings are very large and do not reflect well on your local utility's ability to provide you reliable source of electricity.

Voltage swings like this are likely accompanied by other imperfections such as noise. 

You are a prime candidate for a power regeneration (rectification/inversion) unit. 

@jasonbourne52  Erik's statement is correct.  Lower gauge wire has a higher number.  So if I have a number 12 wire and want to go to a higher gauge, I will go to a 10, not a 14.

@fuzztone 

Indeed, the voltage is not important to most ss amps.  however, the noise and other things that challenge your amp to produce quality sound also are likely increased with poor voltage supply.

many voltage meters can have a zero offset, but if he is using the same meter for all measurements, he observed spread of 13 volts is probably real.

Tube amps can benefit from constant voltage, some more than others.  My power regenerator provides 119.9 volts 24/7/365 so I never have to think about it.

Jerry