moving house: keep same acct or start as new customer?

I will move house soon. I am satisfied with current broadband provider, and I am no longer under contract.  If I want to stay with current broadband provider, do I have to stay on my current account with them, or can I apply for service at new address as a new customer?  A new customer can get one of the advertised deals; an existing customer will have to haggle (ugh!) 
Also:  I want 2 or 3 days of overlap whilst I move.  As a new customer at the new address, I can pick any start date I wish.  Can I have overlap if I stay with current provider and just tell them I am moving house? 

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