You may find this useful. Remember that it is one thing to have the law on your side... it is wholly another to get a NH judge to side for the tenant against the interests of a landlord. We have no protection and few rights in a state where the landlords are a powerful lobby supporting republican interests.

(Note: I am not a Lawyer. And someone who claims this statute doesn't apply to you may not be one either, or may not have your interests at heart.. A first consultation with a legal professional is free. You may want to try that, or you may want to contact Legal Aid and/or try googling for tenant advocacy groups. For more free entertainment perusing NH Statutes, browse to https://www.gencourt.state.nh.us/rsa/html/NHTOC/NHTOC-LV.htm where you will see many of the laws benefitting landlords and very few in defense of tenants. You may also want to keep in mind the following as you study up - Section 540:28:
540:28 Lease Provisions. – No lease or rental agreement, oral or written, shall contain any provision by which a tenant waives any of his rights under this chapter, and any such waiver shall be null and void.
Source. 1985, 244:9, eff. Aug. 6, 1985 )

https://www.gencourt.state.nh.us/rsa/html/LV/540/540-11.htm

(OP, see See Section I.)

TITLE LV

PROCEEDINGS IN SPECIAL CASES

CHAPTER 540

ACTIONS AGAINST TENANTS

Section 540:11

540:11 Termination by Lessee. –

I. A lessee may terminate his or her lease by notice in writing, in the same manner as the lessor, and the notice shall have the same effect for all purposes as a notice by the lessor to the lessee.

II. A tenancy at will, from month to month, may be terminated by the lessee upon 30 days notice; provided that if the date of termination given in the notice does not coincide with the rent due date, the lessee is responsible for the rent for the entire month in which the notice expires, up to the next rent due date, unless the terms of the lease provide otherwise.

Source. RS 209:6. CS 222:6. GS 231:6. GL 250:6. PS 246:6. PL 357:11. RL 413:11. 2018, 18:1, eff. July 14, 2018.