Should I remove existing subfloor sheathing before increasing height of the floor system?
You've asked one specific question here, and I'll start by answering that: No. Unless you have a particular reason to do so, do not bother removing the existing OSB. 

From there I'd say your plan is reasonable. Running the 2x4s perpendicular should pose no problem, as they're supported every 2 feet. They'll span that with no detectable sag. Be sure to butt join them over joists.

Rather than two expensive layers of 3/4" OSB, though, I'd rip 2x6 lumber to the appropriate height so that you can use just one layer of t&amp;g OSB as the new subfloor. This mitigates the overload risk, too.

Note: I assume that you've had the roof system inspected for use as a living space (or even for storage). With 2x8 bottom chords I assume that either 1) it's a very long span, or 2) it was designed for use as such. That's a critical point here.