To settle a disagreement, is the following sentence grammatically correct?
It's slowing my paying my debts off.
Yes, the sentence is grammatical, if somewhat awkward. Out of context, you need both instances of my to make it clear that it is you who is doing the paying off and that the debts are yours.
The words my paying constitute a gerund (paying) modified by a possessive (my). Less formal English would accept the pronoun me here: It's slowing me paying my debts off, which has the advantage of avoiding the repetition of my.
Going by what you've written, you've got an 'it' which refers to something which is the cause of the problem, and you've got something which is being affected ('you paying your debts off'), and the action going from one to the other ('slowing').
In this case, I'd advise 'slowing down', and greater clarity in terms of exactly what is being slowed down.
'Paying off' usually implies repayment in full. So here, 'slowing (down) paying off' sounds strange. 'Slowing down a rate of repayment' makes more sense.
I'd probably advise rephrasing completely - you could say, 'it's ... interfering with/getting in the way of/hampering ... my ability to/my being able to pay off my debt by December' or something like that.