Normally the following applies:
Quote Quoting What if a tenant moves out without giving proper notice?
The law says the tenant is liable for rent for the lesser of: 30 days from the day the next rent is due, or 30 days from the day the landlord learns the tenant has moved out. However, the landlord has a duty to try and find a new renter. If the dwelling is rented before the end of the 30 days, the former tenant must pay only until the new tenant begins paying rent.
Your situation doesn't seem to fall squarely under that provision, as you haven't moved out and you did give notice, albeit notice that the landlord contends is inadequate. (That provision seems to be about a tenant who gives no notice.) Normally, as you had a one year lease (apparently with no automatic renewal absent your giving of notice), your obligation would end at the conclusion of the lease. Had you stayed and paid rent monthly you would have become a month-to-month tenant. Your situation is complicated by the fact that you would be more of a holdover tenant, but you would still owe rent past the conclusion of the lease term.

Based upon what you have said, if the lease was dated Feb. 2 and the premises were available on that date, (or possibly if the lease specified a start or move-in date of Feb 1), the lease started on that date. As you are staying past the conclusion of the lease, it would appear that you would owe twenty days rent from the day you gave notice. As you have not moved out, even if your prior notice was somehow "improper" it would seem that you would still be able to give proper notice and thus avoid paying more than 20 days rent. You can review the state statutes to see if you can find a more specific provision which applies to your situation.