I’ve always had the thought that I would notify my employer that “my circumstances have changed and I’d prefer to stay on as a ‘contract staffing employee’ for the short term to assist in any transition efforts”.
Slow transition is better than telling your boss (i) to pound sand, (ii) to piss up a rope and (iii) your wife is lousy in bed.
Personally, I agree. The process of getting all your legal stuff in place can take time (and you need to do it before cashing in the ticket), so I'd keep working and give notice with enough lead time to allow my boss to back-fill me properly.
Also, with school-age kids, I'd probably wait until the end of the school year to claim, and not even tell the kids until then, so as to not disrupt them mid-year.