"I get started the minute I get out of bed, and I have all the context there in my head, so I'm productive immediately."

Well there's your problem! Get some of it written down, on paper, Evernote, or wherever works for you. You will definitely improve your context switching if whenever you need to switch gears you can review your notes from the last time you were working on that task.

I work as a freelance consultant for several clients, and I keep different notebooks for each one. I don't think I'd be able to manage this if I tried to keep everything in my head.