Your federal refund cannot be offset by this bill. Maybe your state, but it would depend on your state laws. Also, if you were married and filed jointly when the expense was incurred, your spouse could not claim "injured spouse" to get out of it. Only if the bill was incurred before you married, or you were married but filed separately that year.
Filing separately would prevent the state from collecting on the debt IF you were unmarried (or married to someone else) when you incurred the debt, but would not stop the state from collecting the debt if you were married when you incurred the debt. Filing separately usually results in higher tax, because many credits or deductions are disallowed or reduced. Even assuming this debt can be collected via your state tax refund, it does not seem to make sense to file separately for the rest of your life (paying higher taxes every year in most cases) just to avoid paying it.
I would contact the hospital and try and negotiate a settlement for a reduced amount.
