Independent Publishing as a University Affiliate
First, you don't need university affiliation to publish papers. Different universities might feel differently about your situation, so you should ask someone, perhaps in the department for which you teach. They might not want you to use the university's name if you aren't on the payroll, especially if you aren't subject to their normal rules. 

The one important thing that the university might be able to do for you is provide IRB advice if your research involves human subjects in any way. 

Other than that, you can probably use anything that is merely descriptive, such as "adjunct instructor at H*d University". 

But, "Independent Researcher" may be your best "affiliation". The reviewers will be interested what you write in your papers, not your university. Your email address will indicate some association in any case.