How to fix a frequency shifted audio part of an RF signal?
The problem is in using different standards for sending Video+Audio signal over RF. Basically any analog system uses the first carrier frequency to transmit video part and the second (shifted) carrier frequency to transmit the audio part.

I am not good to comment on particular standards; you can find this information in Internet. What you see looks like the frequency spacing between 1st carrier and the 2nd carrier are different in TX part (old gaming system) and RX part (TV). So you can get one (video or audio) component clear, but not two at the same time.

It is very difficult to fix by self-made means.

The easiest way is to dig into menu of your new TV set. May be it supports the standard you need. Then you are lucky.

Alternatively, you can look for a kind of media converter to convert the old standard to some supported by your TV set.