Mama, I am sorry! I am very prone to clogs but I never had to deal with so many so often as you do. I think you are right that there might be an issue with inflammation going on, but I am not sure what you can do about it. My suggestions are palliatives in addition to all the good tips you already got above.
First, I agree on the bra. I never could wear nursing bras or tanks because they gave me clogs, but I also couldn't go braless. So I wear these sports bras:
They don't put pressure anywhere (for me, even seams could cause a clog) and support like a gentle hug.
Not just bras, but even tight-fitting tops and sweaters could give me plugs. I got a plug from the seams in the armholes of my pajamas digging into my underarm area.
Agree with the comment on sleep position. I have to sleep on my side, hugging a soft fluffy pillow for light support. Sleeping on my back causes the breasts to put pressure on themselves (and sleeping on my tummy, needless to say, crushes them).
Do you ever feel like you've fully drained your breasts? I only ever felt fully drained after I controlled my oversupply (that was after 9 or 10 months for me!). When I felt this for the first time, what a relief! It "flushed out" my ducts! I did feel that the constantly full ducts made my breasts tender and slightly inflamed although not as intense compared to what you described, and this tenderness only subsided when the supply adjusted.
Also, do you have lots of letdowns in between feedings? I had those a lot and they contributed to the overfilling and irritation of my breasts. There was also a time when my baby would start nursing, give me a letdown, and then stop before really drinking anything (she was beginning to drop feedings but kept nursing often for comfort).
Oh--I also noticed that stress could spark clogs. Something about the stress hormones perhaps? If I were particularly angry, nervous, or anxious, I would get a clog.
Good luck, mama! Clogs are tough. I hope you can figure this out and get some relief.