I have a question regarding owing back taxes and a divorce

My divorce was finalized in April of 2017. We currently still owe the IRS $600. We are on a payment plan but the plan won't cover what we owe. First, since we were married for part of 2017, will we file taxes as single/head of household (since we have 2 children together and each claim 1), or do we file as married filing separately? Also, our marital settlement agreement says we are both responsible for our debt debt equally. Last year our small refund was applied to our bill. I'm assuming the same will happen when we file our 2017 return. How will the IRS take form our returns? Will they just split the bill between the both of us, or will one return be withheld?