In US English, "as an old local saying goes..." is fine. There is a possible stylistic problem with "As an old local saying says..." The repetition in "saying says" seems awkward to me. You could say "As an old local proverb says..." There might be a slight difference in meaning between "goes" and "says." "Says" suggests an accurate repetition, of the exact words, of a saying whose words never change. "Goes" suggests that the saying varies a bit, and that you are giving the gist of it but not necessarily giving it exactly. For example: "A famous quotation from Shakespeare says 'What's in a name? That which we call a rose By any other name would smell as sweet." "As the saying goes, a rose smells just as sweet no matter what we call it."

No, the expression is ‘as the saying goes’.

Yes, "As the old says goes". British English. Cannot comment for the Americans.

As the saying goes "nothing ventered, nothing gained".

Thanks Luis