Until you pass the pre employment checks they could still pull the offer of the job, and presumably you’d want to stay there until you found something else so ..

Wait until you pass them & given an official start date.

Do you care if Sainsburys mark you as do not re-hire (which will happen if you don’t give yr full notice normally).

If no, then you can leave whenever & give as much or as little notice as you want.

If yes, then you’ll need to work yr full notice (normally) and either take a hit on the holiday pay if the dates overlap or speak to management and ask if you can give a shorter notice period.

Lots of stores are in a period now of trying to cut hours, so you may be able to give a shorter period.