Since you're married, you do need to file either as Married Filing Jointly (MFJ) or Married Filing Separately (MFS) unless you have dependents other than your spouse that qualify you as Head of Household (see more about that here: Do I Qualify to file Head of Household. See also: U.S. Citizens and Resident Aliens Abroad - Head of Household.)
The default election is Married Filing Separately, which doesn't require your spouse to have an ITIN or SSN, but applying for one will allow you to file electronically in the future.
If you choose to file using the MFJ status, you must apply for the ITIN, and you must also attach a statement to your tax return, signed by each of you that states that one of you is a U.S. citizen and the other is a nonresident alien and that you are electing to both be treated as U.S. residents for tax purposes. Include the full name and address of each spouse on this letter.
Include your social security number on the letter and a notation that your spouse has applied for her ITIN.
This election does not affect the immigration status of your spouse, but it will require that you report any worldwide income that she may have on your tax return - such as interest, dividends, etc. Once the election is made to treat a nonresident alien as a resident for tax purposes, this election stays in effect until it is suspended or revoked.
Also, it is a once in a lifetime election. Once revoked, the nonresident alien spouse cannot elect to be treated as a resident in the future, even if married to someone else.
If you wish to file MFS without applying for an ITIN, you will need to ignore errors in TurboTax or fill in a "dummy" number for your spouse and then replace it with NRA (for nonresident alien) on the printed form prior to mailing. You would use this same process for state tax returns.
If you wish to file MFJ, you will also ignore the errors in TurboTax about the missing social security number and print your return. Mail your federal tax return to the address provided in the Form W-7 instructions and not the address that prints on the TurboTax cover sheet.
Since you lived in two different states, you will need to file in each state if the state has an income tax.
You should file an extension (where required) for your state taxes and wait to file until you receive the ITIN back from the IRS. Do not mail your state tax returns with your federal tax returns or they will not be processed.
If you do not have your ITIN before the extension deadline for your state, I would contact them for instructions on how to proceed. Some states will allow an additional extension and others may prefer that you mail in the forms without the information.
There is no reason that you can't do this one on your own, I'm so glad you posted this question!
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