Do I need to tell my old council I'm moving?

Yes. 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), then ask 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.