You can become a psych nurse with a ADN. However, specific facilities may want/prefer BSNs, but that's facility preference and independent of psych. Being enrolled in a RN-BSN program while applying may help you break into those facilities that are strictly looking for BSNs.
I'd start by looking up local psych facilities in your area and applying for jobs.
I don't think you really need a refresher course, as most of the skills you use in psych are not the technical skills such as Foleys, IVs, etc. But definitely brush up on therapeutic communication, common psychiatric illnesses/disorders, common psych meds and side effects. Also brush up on basic nursing skills (VS, etc.), assessments, and common comorbidities such as HTN, diabetes, COPD, the hepatitises (hepatiti?), etc., as you will see them in psych patients.
Consider doing some psych CEUs. You can find a lot of then online for free/low cost. Reading a psych journal or two couldn't hurt: my favorite is the Journal of Psychosocial Nursing. You can also consider joining the APNA (psych nurses association)--it offers educational opportunities, another journal (Journal of the APNA) and networking/mentoring opportunities.