If you went to a psychologist, they don't/can't medicate. If you went to a psychiatrist, they can but if you're opposed they won't. However, when it comes to ADHD, especially if it's severe and affects all areas of her life (ie. school, home, away from home, relationships, interactions, simple tasks, difficult tasks, etc) then you might not be doing her any favors by refusing to even consider medications.
I had the very same mindset as you, and I held out for the longest time with my autistic daughter with her ADD. It was a mistake. She should have been on the seizure med and ADD med much sooner than I allowed it because I was so insistent that MY child wouldn't be on medication. Then I was just as resistant with my middle daughter's ADHD.
So yes, I have a daughter that has ADHD and part of ADHD is anxiety. They go hand in hand. You don't "not raise your voice." You don't walk on eggshells. You may have to be more strict in your parenting and make your guidelines and rules more clear and explicit to the point of posting them on your fridge and include consequences for breaking the rules.
Keep a chore chart on the fridge of responsibilities that she's responsible for.
When you need her to do something, break it down into steps she can follow.
Use positive reinforcement. Teach her organizational skills, but don't be surprised if you have to teach, teach, teach, and re-teach. For years.
But what's made her receptive to therapies and I'll tell you has also improved our relationship beyond what I ever thought possible is putting her on a low dose of ADHD medication. She's still active, awake, brilliant, bright, pulling honor roll in school, smiling, has her anxiety under control, and it's all around been such a positive experience for her that she THANKED me.
We also have reduced/eliminated high fructose corn syrup, artificial food dyes, most processed foods, and some other trigger foods with good results in helping reduce symptoms.
We have an IEP set up in school for my autistic child and a 504 set up with a behavior plan for my child with ADHD (in school).