If you have give proper termination notice, then the tenancy and housing authority contract is terminated upon the expiration of the termination notice. Termination of Section 8 tenancies and housing contracts typically require 90 days notice, so you may have given proper notice. If this is the case, then no further notice is needed. The tenancy was terminated and an unlawful detainer action may be filed immediately.
Please note that if you have been accepting or demanding rent payments or housing authority payments for any period of time after the expiration of the termination notice, then you may have waived the 90 day termination notice. If this is the case, then you may consider serving a 3-day notice to pay rent or quit.
I agree that a timeline is needed to accurately answer your question. If you have accepted rent etc after the notice, then you need to file a new notice - probably a 3 day notice to pay or quit and then proceed with the unlawful detainer process if they still do not move.
If they have not paid you any rent past the 90 day notice, then you should be able to proceed now with the UD based on your 90 day notice.
This answer is a general interpretation of the law and is not fact specific to your case. Likewise it does not create an attorney-client relationship. You should seek an attorney for a review of your specific facts and documents. My firm is All for the Family Legal Clinic, Inc. a nonprofit public benefit corporation that charges on a sliding scale based on income and family size. For a consultation contact us 510-999-7732 or at our website www.allforthefamilylegalclinic.org