The felony record is going to make this a more involved process than a standard application, but it is not automatically disqualifying, and you have two significant factors working in your favor. Five years clean with no new charges shows a track record of compliance since release, and being the mother of his children gives the warden's office a concrete, child-centered reason to approve the visits.
Arizona State Prison Complex facilities understand that children need access to incarcerated parents, and the administration is generally motivated to find a way to make that happen when the request involves a parent bringing minor children. The children's interests carry real weight in that evaluation.
What matters most is how you approach the application. Be completely transparent about your record. Do not minimize or omit anything because they will know exactly what your history is. Come across as humble, respectful, and focused on the children's wellbeing rather than making demands. The moment the application feels entitled or adversarial, the path gets harder. The warden's office has discretion in these decisions and they use it in both directions.
Submit the application, attach any documentation that supports your stability since release, and be patient. This may take longer than a standard approval and may involve additional review. Engage with the process professionally and give them no reason to deny it.