You don’t need a new Apple ID.
Leave the Family Sharing group:
Leave Family Sharing - Apple Support
set up your own billing on your existing Apple ID:
Subscriptions and Billing - Official Apple Support
and use Safety Check:
Personal Safety User Guide - Apple Support (there’s a PDF of that guide at the bottom of that webpage)
and review your trusted numbers, and enable two-factor authentication if not already:
Manage two-factor authentication for your Apple ID from iPhone - Apple Support
Family Sharing does not share passwords, though you might want to consider whether any of those passwords might be or are known, and preemptively change those.
Why not new? Swapping to a new Apple ID loses all of your context, including playlists, app purchases, iCloud data, saved passwords, etc.