Yeah this is straight though Three, my current contract was with Three through a third-party site
Can I do that again to go with Three, or would I have to change provider as well?
Yeah your better off starting as a new customer if Three offer you nothing.
Just get your PAC code when cancelling and its live for 30 days and you can switch to whatever network you want (even go back to Three) includes SIM free or PAYG as well just give the new network your code and you get to keep your old number.
I'm in the same boat as you but to me my Z1 still works perfectly and see no reason to upgrade yet, probably wait for Sony's new flagship/Nexus 2015.