I found my position just by applying on their job board - azstatejobs.gov. There are so many openings and all of the different position titles were overwhelming. I just searched by keyword and read each description until I found a few I was qualified for. I submitted my resume online and was called about a month later.
The call was sort of a phone screen. The manager asked me a few questions to gauge my understanding of the job. Then she invited me to an in-person panel interview (there were 3 on the panel). I showed up 15 minutes early, brought copies of my resume, and dressed professionally. Prior to the interview, I did some research online about the agency that was interviewing me. There are so many agencies in State government and they each have unique missions - I think this was key to showing my enthusiasm for the job.
They had me take a computer-based test right after the interview. I never found out my score, but I was offered the job so it must have been ok.
TL,DR: Applying for jobs online feels like sending your resume into a black hole. Be persistant, sometimes it works out.