It depends why you can't take them. I can't take NSAIDs, because they're contraindicated with one of my prescription meds. That means no aspirin, ibuprofen, naproxen, diclofenac, mefenamic acid, etc. This has been very much on my mind lately, as somehow over Easter weekend, I did something to my arm and there was great pain and very little arm movement.
I took cocodamol for the pain, and saw a joints man at the end of last week, since when I have also iced the shoulder for 10 minutes twice a day to help with the inflammation, and have exercises to help rebuild strength and movement. I knew that with such an injury, time would be a big factor, but I also didn't want yo be doing anything to exacerbate it, and I don't want a repeat.
If you are not taking them because of past reactions to ibuprofen and naproxen, I'd have thought that will slso be a risk with other types, but they might be worth trying anyway.
It's worth seeing the experts though.