But your situation is very common, your current school will have had this countless times in the past.
Keep them updated, don't hand in anything written yet and as soon as you have the signed contract resign.
They've already advised that they want it in by today, I presume they want to be in a position to appoint my replacement asap, they can't really appoint until I've resigned as they'd potentially be doubling up on staff, as I understand it anyway?