What could have happened as well is, the wire going to the switch (On your shifter) has a short in it. Try putting in a new wire that just might clear it up. After shifting for a long time you keep bending and moving that wire so naturally it develops a short in it.
If it is the switch on your shifter SHOP! GM wants 150$ Canadian for it.
As Red suggested, check your wiring first. If that wiring is ok, the switch for the overdrive button is located on the front of the shifter handle at its base, just under the boot.
It's been a while since I replaced mine, but I'll try to cover all of the bases. If I missed anything, someone else will chime in and correct me hopefully.
Remove each of the trim plates (the driver side, the info center plate and the console plate) so you can lift the boot up.
Gently pry the button from the top of the shifter.
Unscrew the T-rod from the shifter handle. Be sure to count the approximate number of turns it takes to remove so you can more easily find where to return the T-rod when you reinstall it.
Remove the shifter knob.
Loosen the nut that secures the switch to the shifter and then remove the switch and actuator block from the shifter handle. Be careful not to loose the return spring located inside the actuator block.
Disconnect electrical lead at the switch and remove the switch and actuator assembly. The two pieces can be separated by removing the pin which holds them together.