Primary docs typically aren't trained on psych meds. Sometimes you'll find one that is. That being said, a psychiatrist would do the same thing as a regular doctor: prescribe meds, see how it works, change it if it doesn't. It's just that psychiatrists know what to ask and know symptomatology better.
I don't know if you'd qualify for Medicaid...that's a whole process, but if you go through the CMH in the county you live in they can help with that.
There are better ADD meds than Adderall: Concerta, Vyvanse...others I'm blanking on right now. Yes, there is potential for addiction, but that doesn't necessarily mean you'll become addicted.
My personal experience with meds: I was on Wellbutrin. It made my migraines worse and I felt numb emotionally, so I stopped taking it. When I was in the hospital for attempting suicide in 2015, I was put on Effexor. I still take it. It has been a literal life saver. I know I wouldn't be here today if I wasn't still taking it. Do I still have days when I feel sad? Absolutely. That's life, and I work through it. It's not the crippling depression I had before.
Taking psych meds does not make you weak. It takes a lot of strength to say " I need help," and take the steps to get help.
Going off meds is a process of titrating down until you're off of them.