2003 chevy cavalier shifting problems

run the fault codes
check for torque converter clutch servo/solenoid problems

Backup Lamps
The B/U LMP fuse #42 in the underhood junction block supplies ignition positive voltage to the park/neutral position switch. When the driver places the gear selector lever in the REVERSE position, the current flow is from the park/neutral position switch to the backup lamps. The backup lamps are grounded at G401 and G402.
Are you also having shifting problems ? If not , I can tell you it's not the fuse ! Wiring problem, the bulbs , park/neutral position switch !
But

Sounds like the reverse band or servo has gone out. Transmission shop needed for a confirmation.

It should start in neutral position. Neutral safety switch is usually located under the shift console by the shift lever.

The neutral safety switch will not let you crank the engine in anything but park and neutral. If something happens to the shift cable the trans may still be in drive.

Manual or automatic?

Had the same problem. to shift gear manually with your hands. Look under the brake fluid (about a foot down) and there is a flat piece of metal. On the end of it on the bottom is a little rod with a round end. That is where the shift cable connects. pushing it toward the driver side of the car will put it in park. And if you count the number of clicks from park you can change gears. For example: 1 click from park is Reverse, 2 clicks is Neutral, 3 clicks is Drive. That is if it is all the way towards the driver side. Don't force it. when it stops clicking dont push it anymore. Hope this helps.

linkage maybe

theres a shift cable if its floor model,linkage for column model.

try this :
with the car in park start it, then from park move it down one. if it is reverse move it down another one, now that should be neutral, then the next one should always be drive