Disassemble it a piece at a time.

I would peel off the trim and drywall, see what you've got and cut it into pieces. Is it finished on top or just exposed framing?

I would do as Verbo suggested and drywall it in up to the ceiling instead and keep it.

I agree take it down. Adding up to the ceiling would only involve more framing and drywall weight that would need additional support. Score the drywall where it meet the wall to break the tape lines. This will help to keep the drywall on the walls from most damage. Remove the drywall from the frame. Then depending on how the framework was built disassemble it. Then you will have to patch the drywall where the frame was. Due to the texture on your walls, get a professional to patch. Especially if you never done drywall before.