If I understand you correctly, you switched from an employer who provided you with coverage in an HDHP (High Deductible Health Plan) to an employer who did not provide such coverage. In TurboTax when you entered the number of months you had HDHP coverage, if it was not for the whole year then your annual HSA contribution limit is reduced.
OK so far?
You are saying that TurboTax is reporting an excess contribution of $350, which is certainly possible in this situation.
In your case, I assume that the HSA value on December 31 of your current tax year was zero. The system will assess a 6% excise tax on the lesser of (1) your excess contribution amount carried over or (2) the value of your HSA on December 31 of your current tax year.
If your HSA value is now zero, your penalty will also be zero.
If you ever qualify for an HSA again (that is, get HDHP coverage), then the carryover will be applied to that first year. It will be as if you made a personal contribution of $350. Remember, to allow for this when you are making HSA contributions again in that first year.
[Edited 3/17/2020 3:57 pm - made year neutral]
