If you are married and living together you can only file as Married Filing Jointly or Married Filing Separately. Even if you are not living together and are legally married you cannot file as Single.
See this TurboTax support FAQ for the Head of Household filing status - https://ttlc.intuit.com/questions/2567672-can-a-married-person-claim-head-of-household-filing-status
Whomever pays for the health insurance is not relevant to the filing status on a tax return.
