Do I have to pay for Post separation joint debt?

It is not a real thing as you describe it. Normally debt incurred by a party post separation is the sole responsibility of the party incurring it. However there is such a thing as Occupational Rent – where a spouse has been ousted (forced to leave) a matrimonial home, that ousted spouse can claim compensation for 50% of the rental value of the home that he was kicked out of and that his spouse continues to occupy. Of course that occupying spouse can counterclaim for compensation for half of the property taxes, house insurance and the principal portion of the monthly mortgage that they continued to incur after the ousted spouse left. Usually these countervailing claims offset each other – or judges may say these claims cancel each other out regardless of the precise mathematical calculations having an offset.