I have to take my cat on a plane but don't know exactly how to go about it. Does anyone have advice?

Woozy and sedation are basically the same thing. There is nothing safe that can be used to reliably sedate your cat that will not affect its blood pressure, ability to keep warm, etc. If it is a short trip (2-4 hours or so) he should be fine in the cargo hold, so long as you are not going in the dead of winter or to someplace like the Arctic. The absolute safest way to transport your cat is by car. Those tiny carriers designed to fit under an airline seat are not really big enough for a mature cat and definitely not comfortable either. I would talk to my vet about maybe getting the cat on some anti-anxiety medication to start well before the trip since he is over grooming, that might also be useful for the trip itself. You will have to have a health certificate issued by the vet for the airline anyway, so he will need a physical. Call the airline and see how far in advance you can get the health certificate, it varies depending on where you are travelling to.