I used to work in Council Tax and this is definitely one for the billing side of things, so council tax. They will try to fob you with the whole it has been passed to the bailiff bit but you are disputing the bill.
Your first bill for £650 seems ridiculously high for a 2/3 month period.
I would email them, you should get a standard "we will deal with your query within X number of days bit"
Ask them for a breakdown of the charges, from when and to when. This is a very simple task for them. They can then also list the payments you have made against the account.
I personally think something has gone very wrong with the billing side of things. When a bill is produced mid year it would say "charges from 1/11/11 to 31/3/12.
Then the next year's bill would be 1/4/12 - 31/3/13 even if you had called them to tell them you were moving out on 1st May (I know you didn't) but the bill is always apportioned after the move date in case something goes wrong with the move.
So your second bill should say 1/4/12 - Sept 2012.
Any discounts (single person/occupancy) should show and the payments should be listed. Honestly, it is the most simple thing for them to show you. If there is any query over where that other bill came from then the person who dealt with your call is at fault for not dealing with it, and investigating why you have another amount outstanding.
Sorry this is long, I am also very concerned how fast the first part of the bill was escalated to bailiff stage. There is legislation laid down about how you are billed, then reminded then final billed, then court date for a liability order, then the court date and liability order produced after that date and then follow up. Seems remarkably fast.
Could the first bill be from your previous address?
At the end of the day, email the council, state you are disputing the amount, are happy to pay anything you genuinely owe, ask for complete bill breakdown and payments made.