Yes — and this is the one people forget, which is why it's also the one that costs the most.
Cancelling your Direct Debit does not close a council tax account. Only the council can do that.
Give your old council your move-out date and a forwarding address. Most have an online moving form, which is quicker than phoning. Then ask them when your responsibility for the property ends and what the closing bill covers — depending on your tenancy or ownership and who moves in after you, you can owe council tax for a period after you've left, and at both homes at once.