Yes, handwritten letters are generally accepted at Arizona State Prison Complex facilities including the Flamenco unit, and standard notebook paper is typically fine.
That said, every unit within the Arizona Department of Corrections has its own specific mail guidelines, and Flamenco houses a specialized population which can mean additional restrictions beyond what a standard unit allows. Before you send anything, it is worth calling the facility directly or checking the ADCRR website to confirm the current mail rules for that specific unit. Rules around paper type, ink color, envelope size, and what is allowed inside the envelope can vary and change without much notice.
A few things that get mail rejected at most Arizona facilities regardless of unit are colored paper, envelopes with any kind of padding or lining, stickers, glitter, lipstick marks, and anything that could conceal contraband. Keep it simple, plain white or standard notebook paper, blue or black ink, a standard envelope with nothing extra inside.
If you want the added convenience of not putting your home address on the envelope, InmateAid can handle the letter for you and send it from their Florida address. That keeps your personal information off mail going into a correctional facility, which is always worth considering.
When in doubt, plain and simple gets through. Anything that raises a question in the mailroom gets rejected.