CHIO my man.. it is carburated .. Yo Nasty your are correct .. lxi is injected dx - lx have carbs ..

KX,, have you checked the pcv valve?? when car idles rough. get some pliers and squeeze the pcv valve hose see if rpm smooth out ...

when you say ( it seems like there is no power to the starter) did you physically chk to see if there is power at starter solenoid ?? if not you need to do that ,,it is the white/black wire at starter
and on the finale note........DO NOT start replacing parts and hope to get lucky untill you properly DIAGNOSE the problem