'as the old saying goes' is correct and can be used in written English. It is generally used to introduce a popular saying, proverb, or idiom. For example: "As the old saying goes, 'there's no such thing as a free lunch'.".
Goals change games, as the old saying goes.
As the old saying goes, "Laughter is the best medicine.
As the old saying goes: You are what you eat.
As the old saying goes, in vino veritas.
What gets measured gets done, as the old saying goes.
And, as the old saying goes, defense wins championships.
As the old saying goes, "A Marine on duty has no friends".
I love the desktop app, it’s always running on my Mac. Ludwig is the best English buddy, it answers my 100 queries per day and stays cool.
Cristina Valenza
Retail Lead Linguist @ Apple Inc.