Will the current drywall look better after I texture it?
That's not exactly a quality finish - by texturing it you're just hiding a bad job.
If time permits, I'd rip off the old tape and most of that plaster/mud, and re-finish this wall in the same way you've done the other new walls.  That way your texturing has the same base to sit on, and more likely to age the same way as the new walls.