My 92 Yukon started doing this just after warranty. I found replaced the 25 amp fuse and as soon as I'd 4WD, it would blow again.
The dealer didn't let on until after I surrendered the vehicle for repair that it was a relatively cheap and easy fix. Front Axle Actuator. Back then it was about $125 GM installed.
As it was explained to me, the actuator, when activated moved a pin similar to an irrigation solenoid pin to move and engage the front axle.
I don't have any history with the Avalanches I've owned. I hope your fuse holds up.