Can the expression "fully booked" be used metaphorically legitimately?

"I'm fully booked." or "I'm booked up." is common usage. A quick look around online suggests that it's common in British English than American but I don't think any of us "Yanks" would be confused or even momentarily surprised to hear it.

Originally, theaters and restaurants were booked and people were just busy but I think we can all be booked up now.

This isn't metaphorical. Booking is an everyday process where people take slots (for example, a superstore arranging to receive food trucks at a certain hour of the day). That's what booking is: arranging to do a thing at a certain time. It isn't "metaphorical" or "figurative" for you to do that with your own time, even if you are a poet, and not a dirty truck.

Yes, your usage is totally acceptable and also very normal and already exists. If you don't like it, then use your artistic skills to invent a new word, or simply say you are "busy".