The biological parents always have first right to claim their child. Since you lived together all year (no matter where that was, or whether it matches your driver's license), the parent with the highest AGI claims the child (tie-breaker rules).
The only way someone else could claim either of you (or your child) is if they 'provided over half of your support' and with your income, that would not be feasible, even if they provided temporary housing for you.
So the parent with the highest will file as Head of Household and claim your child on your return. Click this link for more info on qualifications for head of household.
This link gives more detail on Tie Breaker Rules.
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