Anxo, I have a 1988 K2500 with the floor lever and it is still electrical 4X4 engagement. There is a motor that locks in the transfer case. This is why the 4X4 light dosn't turn off/on immediately when you shift out/in of 4X4 (mine anyway). I would say you have a seized motor or jammed motor (pulling too many amps) or a short. Get under the truck and you'll be able to trace the wiring.
HA, I can relate to this topic. When my brother bought the truck I now own brand new, 92 Chevy 1500, he came home and was trying all the buttons and switches and stuff. For the hell of it, he tried the 4wd, it has the lever on the floor. He put it in and nothing, it didnt light up either. He was like hmmmmm, he checked the fuse and it was popped. Now he was a little mad, he just brought the truck home like 15 minutes ago. Just for the hell of it, he unscrewed that plastic cover, over the 4x4 handle where it goes down through the floor. Theres 4 screws there, on ours anyway. Just by luck when he took the cover off, the knob had to come off too by the way, he noticed wires that were crushed and it was grounding out. He fixed the wires, retaped them, and ran them a slightly different way to avoid getting pinched again. Put in a new fuse, done deal. 4wd worked perfectly after that. Not saying this is your problem, but this is what happened to ours. Right from the factory, the wires were crushed behind that plastic cover. He was happy thats all it was and he was able to fix it in our driveway with barely any tools. Been good ever since. Yours is gonna be some sort of short somewhere too. Mike