There is a shift interlock system on that car which prevents you from shifting out of park until you hit the brake pedal. Basically how it works is there is 12V power at the brake light switch continually. When you touch the brakes, the switch sends power to the rear brake stop lights AND power to the solenoid on the shifter handle at the same time. If the solenoid is bad, it wont release a catch on the shifter. If you have a bad brake light pedal switch, it's not sending power to that solenoid. And here's the kicker. The fuse that powers all this nonsense is the fuse for your turn signals. Start by checking that fuse first. Most likely you have a brake switch on the pedal that is grounding out causing the fuse to blow. The switch is $20 at the auto parts store. The BWD part # is S6072. I know, had the same problem.
If you do not have your car turned on the vehicle will not shift out of park. You also have to have your foot of your brake.
The cable between the shift lever and the transmission may be bad.
Check shift linkage for wear or binding or bent parts
Sounds like something is out of alignment in the transmission area. Make sure all the bolts are tight down there.
Align the shift indicator orreplace it. Your problem is strictly column mechanical.
check the fluid level
You sound like you have the same problem I had. The shift lever has a linkage that overtime can stretch. When you put the car in park, wiggle the shift lever until you hear it click. Then you can remove your key.
are you pressing the brake as you try to shift?
Sometimes when used to driving manual transmissions, we forget to press the brake pedal to release the safety so we can shift the automatic into gear. It will not shift out of park unless you do this.
footbrake deactivation switch
I don't know for sure but on some fords if the fuse for the brake lights is burnt out it will not shift from park. On fords if you turn key part way to on position then shift to neutral and start engine. Do not use park till you get fuse replaced. Mine blew because of bad trailer light harness. Sorry if this does not work on chevy.
vacuum leak?