I have not ever worked on this model,but here's something to try. Disconnect it at the transmission shift linkage on the transmission and see if it will shift by hand. if it will then you will need to look at the cables,rods or the shifter mechanism.Maybe something is worn,bent or broken.
If it is standard, it is the linkage. If auto, the trans needs overhauled.
Could be a linkage problem..
Auto or standard?
Time for a transmission rebuild.
Door : 4x6 Rear : 6x9
Is this a standard transmission, or an automatic? automatic
Begin by removing the 2003 Chevy Cavalier alternator belt. Remove the wiring from the back of the alternator. Remove the alternator retaining bolts. Reverse the process to install your new alternator.
Remove the belt from your 2002 Chevy Cavalier alternator. Remove the wiring harness from the back of the alternator. Remove the alternator retaining bolts. Reverse the process to install your new alternator.
Disconnect the positive cable from your 2003 Chevy Cavalier battery. Remove the cables from the front of the starter. Remove the starter retaining bolts. Reverse the process to install the new starter.
if it is the same size engine all parts should fit.cavaliers from 1995 through 2004 are the same
Door speakers are 4x6 rear is 6x9s
Defective reverse light switch.