You are correct. Per IRS rules, if you were divorced on December 31, 2016, you have two options to file: Single and Head of Household (if there are qualifying dependents). You would be allowed to file as you suggest above in this case. You may claim the home expenses if you paid them. Your ex, however, would not be able to claim Head of Household unless your ex paid for at least half of the household support. Here's an FAQ on this: https://ttlc.intuit.com/replies/3288637
This is the general rule, but there is an exception if your state recognizes common-law marriages. Then, your situation would depend on how state law is interpreted in your case. If this in your state your living situation is deemed a marriage, then you are not allowed to file as single individuals; you must file married (While it may seem odd that you could be considered married since you have divorced, your living together as you are could be interpreted as having remarried in some common-law states. If you wish, you may comment on which state you live in and I can provide you with what the law says if common-law is recognized in your state)
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