I don’t think I’ve ever done that lift experiment on my own 4wd truck. I just assumed all four tires would spin if I tried it. I have limited slip differentials both front and back though.
I grew up with guys who were big into off-roading (mud). Common practice was to “weld the spider gears” together, in both differentials. I’m not an expert on the insides of a differential. I do know this would cause all 4 wheels to be driven by the engine at all times (when in 4wd). And when they eventually got stuck, they were reallly stuck. I used to drive my Jeep around the mudholes they’d attempt to go through. I remember seeing rear bumpers and driveshafts laying around in the mud. Hence, I drove around on the high spots 
. Mud is fun, but I grew out of it. Too much money for repairs and too much of a pain in the rear cleaning all that goo off when the fun’s over.
When not loaded and off the ground, both tires may spin but one will be easily stopped due to the open diff. Stock trucks have open front diff otherwise turning in 4WD would be a nightmare. Great for serious offroading and mudding, not so good for normal driving…
My '88 K2500 had the automatic locking actuator on the front axle (thermal actuator as described above). What’s confusing is there are two versions (at least) a normal and heavy duty and the axles and hubs are different as far as I know. How many lugs on the OP’s truck? First thing to check is the linkage from the shifter to the transfer case. With a lifted truck, that should be easy 
He says he has manual locking hubs.
Correct. The “light duty” 3/4 ton had auto hubs and independent front suspension, I believe. 6 bolts on axle shafts / 6 lug nuts. The regular 3/4 ton and up, I’m not sure but I think still had a solid front axle.
OP’s truck may or may not have the original axles. Is pretty common for folks to swap in a solid front axle in place of the independent front axle.
Also pretty common to swap in the 14 bolt (on diff cover) light duty 3/4 ton rear axle to replace the rear 1/2 ton axle. Have done that one myself. Pretty neat and easy swap.
Either way, if it has manual hubs, pretty easy to check stuff. Like you said, linkage could be off on transfer case. Need to shift to 4wd and try to turn front driveshaft by hand. If you can turn it, transfer case isn’t engaging. Then you can go from there.
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turn steering wheel to left and punch it. you can should see stuff being thrown by tire?
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At the time I posted that the further details that OP provided hadn’t been provided yet.