My page count went up after I set chapters to start on a right-hand page. Will my KDP cover spine still be right?

Yes. From version 1.9.24, the blank pages added by "Start chapters on a right-hand page (print)" (in Book design) count in the book's page count, and the spine width of the KDP cover wrap follows it. For an exact spine, export the PDF after you tick or untick the option and before you make the KDP cover wrap, so the wrap is made from the new page count. Upload that same PDF to KDP as the interior.