Are you sure the hum is from the transmission and not a wheel hub/bearing or the transfer case? You can install a long piece of rubber hose on the transfer case and front diff. vent and listen as you drive. If sound heard thru either one suspect that component. As for the shifting issue, does the engine shift at all or stuck in 3rd gear? GM does have problems where the ignition switch can cause the transmission to stay in 2nd or 3rd gear. Here is a bulletin pertaining to that very matter ok.
#01-07-30-002E: MIL On, Transmission Stuck in Second or Third Gear, IPC Inoperative, DTCs P0740,P0753,P0758,P0785,P0787,P0973,P0976,P1860,P2761,P2764 and/or P2769 Set (Verify and Repair Source of Voltage Loss - (Sep 23, 2005)
This bulletin is being revised to update model years, add additional models and add additional diagnostic information. Please discard Corporate Bulletin Number 01-07-30-002D (Section 07 -- Transmission/Transaxle).
Condition
Some customers may comment on one or more of the following conditions:
Upon investigation, one or all of the diagnostic trouble codes (DTCs) P0740, P0753, P0758, P0785, P0787, P0973, P0976, P1860, P2761, P2764 and P2769 may be set depending on the vehicle.
Cause
The most likely cause of this condition may be a loss of voltage to the transmission solenoid power supply circuit.
This condition may also be the result of an incorrect installation of an aftermarket electronic device such as a remote starter or alarm system.
Correction
Important: If the vehicle is equipped with an aftermarket electronic device and the DTCs are being set when the device is being used, verify that the appropriate fuse is being supplied battery voltage during operation. If voltage is not present at this fuse, these DTCs will be set due to lack of voltage at the solenoids. This condition is most likely to occur with an incorrectly installed remote starting system. If this is the case, refer the customer to the installer of the system for corrections. ANY REPAIRS DUE TO AFTERMARKET INSTALLATION OF ACCESSORIES IS A NON-WARRANTABLE ITEM.
Auto Technician
GM Mark of Excellence/ Service Guild Participant with 13 years experience, ASE Master Certified