You should consult a landlord tenant attorney. If your landlord did not agree with your way of making the last month's rent payment then technically your landlord can send you a notice to quit for nonpayment of rent. It is your landlord's determination to make any deductions for damages to the unit, not yours. That being said, your landlord should have given you a receipt for your security deposit and have held it in an interest bearing account. Failure to follow the security deposit laws can subject your landlord to multiple damages. To avoid any possible problems on your end, you may want to ensure your rent is paid in full and deal with the security deposit issue afterwards. If the unit is immaculate as you say then you likely would not receive any deductions. Further when you get your security deposit back it should include interest earned during your tenancy. Your best option is to sit down with a landlord tenant attorney and discuss what rights and remedies you may have. Best of luck.
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Please be advised, your security deposit is not "last months rent". Therefore, your landlord is not required to agree to your suggested payment arrangement. I would advise consulting a landlord-tenant attorney whom may be able to negotiate a favorable outcome for you, especially where it appears the landlord has not complied with the requirements of the security deposit law.
Justin M. Murphy
I agree with previous responders as to the security deposit and rent. But to answer the question you asked, you don't become a trespasser if you overstay. Landlords and tenants cannot be considered trespassers vis-a-vis each other. If you overstay, the landlord's remedy is to file an eviction case in Housing Court or District Court.
Disclaimer: This site exists to provide information only. It is not legal advice. Answering your question does not create an attorney-client relationship. I am a Massachusetts lawyer. Any information provided on this site does not, except as explicitly stated, imply familiarity with laws or procedures peculiar to your state which may differ from those where I practice.