Migraines tend to be very hereditary. If you have relatives with migraines, chances are that you are having them too. I've had them since I was 8, so that's about 34 years worth. Needless to say, I'm pretty well versed in all things headache-related.
I think you're missing the mark with seeking help via the chiropractor. Chiropractors are like Lays potato chips, you can't go just once. When Chiropractors give you adjustments, they are just temporarily realigning your vertibrae. This is pretty much like when people crack their knuckles or other joints. The noise is basically just squeezing out the air that has accumulated in the joint. It feels better for a little while, but it doesn't last.
Because your spine alignment is determined by the condition of your muscles, your chiropractic treatment gets undone rapidly as your muscles return to their normal state. To get more lasting results, you have to work on your muscle strength.
Personally, I think your Chiropractor sounds like he's full of it. Anti-oxidants aren't going to do a blasted thing for migraine pain. A quick x-ray will show if your sinus cavities are full of gunk, so you should probably go directly to the ER and get some answers. Unless you are blowing green stuff out of your nose, it's probably not a sinus infection.
Your posts don't mention you taking any kind of pain medication. Are you taking anything for the headaches? You also didn't mention why you think TMJ might be a possibility.
I seriously doubt TMJ is the problem, since TMJ pain is mostly in the jaw joints and farther down on your face(your mouth and teeth would hurt). I've been dealing with TMJ on and off for about 24 years, so I'm very well versed on this miserable condition also. TMJ is pain comes from inflamation in the jaw joint. After being on Prednisone for so long, the likelyhood of having any inflamation in your jaw joints is miniscule.
I'm really betting on Migraines being your base problem, and too much medication or taking medication too frequently has thrown you into a cycle of rebound headaches. Again, I'd think I recommend hitting the ER with your headache problems so you can get on to a quick diagnosis. If you try to get a referral to a neurologist, it could take weeks or months to get an appointment.
Let us know what happens, so we can give you any information that might be helpful in resolving your problem. Good Luck!!