This notice sounds confusing. Had the landlord put a date on which your tenancy would end, your signature would have made it enforceable. But it does not appear to be so from your description.
Your landlord probably waived February rent based on the understanding you were moving out quickly. You still owe February rent since you are still there. You also owe rent for all of March. You may be moving out the 16th, but your tenancy ends the 30th.
Your landlord may be willing to accept less rent for March if you move out before the end of the month. See what deal you can come up with. Get a signed agreement stating, for example, that you will pay half rent and move out by the 16th.