If you have a valid common law marriage (this is separate and complicated discussion) then you file a single tax return as "married filing jointly." You are essentially one taxpayer, and you file one return that lists your combined income, deductions, credits and dependents. So you and your spouse (not your partner, they are a legal spouse if you have a common law marriage) can take any valid tax deduction, regardless of whose bank account the money came out of.
As noted, moving expenses are not allowed on a federal tax return unless you or your spouse are in the military. But you can enter moving expenses because they might be deductible in your state, and after you list them on the federal side of the program they flow down to the state side.
If you are filing separate returns, you must file as "married filing separately," and not as single. In that case, each spouse reports their own separate income, but deductions can be divided in any way you choose. It is more conventional for the person who paid the expense to deduct it, but it can be treated as though you gave money to your spouse and your spouse paid the deductible expense.
You can't file as single, or head of household if you lived together. You are legally married. If you want to discuss further the concept of common law marriage as it applies to tax returns, let us know.
