Wow, you have bee suffering with this for a long time.
First, I was concerned by your description of "with circular places that looked sort of like ringworm" because that seemed like a highly specific reaction. So I looked, and antibiotics can cause something called Stevens-Johnson Syndrome
https://www.drugs.com/cg/stevens-johnson-syndrome.html which is described as :

"Stevens-Johnson syndrome (SJS) is a rare and serious condition of your skin and mucus membranes. SJS will cause you to lose up to 10% of your outer layer of skin. SJS is usually caused by a response to a medicine you have been taking. The most common medicines are antibiotics, NSAIDs, and antiseizure medicines. The response may happen 1 week to 2 months after you take the medicine. SJS may also be caused by infection, vaccinations, or diseases involving your organs or whole body.
You may have a fever and chills up to 2 weeks before you have skin symptoms. You may also have a cough and sore throat, headache and body aches, and feel more tired than usual. Skin symptoms include the following:
Sores that look like targets
Painful mouth sores that make it hard to swallow or breathe
Painful blisters on your skin, eyes, or genitals
Sunburned appearing skin once the top layer falls off
Your healthcare provider may diagnose SJS when he sees your skin. Tell your healthcare provider if you have been taking any medicines recently. A sample of your skin may be taken and sent for tests to check for SJS."

When you were being tested, did the doctor do a thorough history of what medications you are using? Did they check to see if any other medications or supplements you are using might be interacting with the antibiotics?

There are other causes of rash while using antibiotics including:
Azithromycin lists "blistering, crusting, irritation, itching, or reddening of the skin" as less common side effects
Ethambutol lists "skin rash" as a rare side effect
Rifampin says if you "develop a skin rash, hives, or any allergic reaction to this medicine, check with your doctor right away."

At this point, if I had the rashes you describe, I would first call my pulmonologist (and ID doc) - they would be most likely to be familiar with these reactions. Then I would try to get all the doctors talking to one another to figure this out.

Can you initiate a phone call to you pulmonologist to describe what is happening to you?

The doctor treating you for the MAC should do all the tests
They should be on alert for any adverse reactions. When presented myself with rashes the response I got was “oh you’ve got thin skin”