C shaped, deep red/purple rash to bil armpits. Has "rope burn' appearance-poss staph/strep inf.- extremely painful -how to relieve discomfort?

Rule out HS. Trust it's not deodorant allergy. The entity to rule out is hidradenitis suppurativa. A physician's office visit is in order and then perhaps a dermatology consult.
Thoughts.. This sounds like a really concerning rash. I hope you have gotten it checked out! if it's a cellulitis or skin infection then antibiotics would be the number one thing to make the pain go away. Specific pain medicines like Ibuprofen and tylenol (acetaminophen) could also help. But the distribution of the rash you describe is very curious, so close followup with your doctor is important. Best of luck!