I used to work in council tax
You can give your details to the council about the date you are moving, who is responsible for your property once you leave, and the new property you are moving to.
As you are moving within the same area leave your DD set up and they can transfer it. They should only process the information after the date you leave in case of any problems with the move. As they won't get to do this until February the first payment they can ask you for is 1st March.
If your new house has the same band, you won't pay anything. If you are moving up a band there would only be a small amount to pay. Have a look at the cost difference between your current band the house you are moving to. If you don't know it and even if you have been told it check it here
If the new property has been extended it would tell you on the VOA site (link above) if it is pending a re-band. I did have a very upset customer once who thought their council tax would be a C and it was re-banded to an E which where I worked was a hell of a jump.
Every month you are in a property, you pay for that month in advance plus a bit extra to cover those two "free" months. Even if a person knows they are moving in May the council has to assume that the bill covers the entire year, so they would bill a full year, So 1st April to the 31st March the following year. Then in May, they apportion the bill back to cover only April to May. Hope that helps.