If you are divorced as of December 31, you must file as divorced, you are not married for part of the year. You will file as single or as head of household if you have a qualifying dependent and meet the other tests.
If the debt is from joint returns you filed together when married, the IRS can collect from either spouse even after the divorce. The IRS can collect from whomever has money, they don't have to collect 50/50 or honor your divorce agreement.
However, if the IRS does collect more from you than you think is your "fair share", you can file a claim for separation of liability or equitable relief asking the IRS to divide the debt equally. https://www.irs.gov/businesses/small-businesses-self-employed/three-types-of-relief
If you end up paying all the debt and the IRS does not grant a request for separation of liability or equitable relief, you would have to pursue your ex in the family court to recover his share of the debt from him.
