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Did you try all the normal things like popping it into neutral allowing the vehicle to roll slightly and engages into four-wheel-drive? Are all of the tires worn evenly for our to the tires newer than the rest?
If the lights are flashing then the system is not engaging four-wheel-drive. The problem is usually never the transfer case itself. 90% of the time it ends up being the actual actuator that amounts to the back of the transfer case. What I have done in some cases is climb underneath and the rubber mallet bang on the actual transfer case actuator carefully while someone tries to engage the four-wheel drive. Block the wheel so that you don't hurt anyone. Or ideally put it on a lift. Sometimes jarring the transfer case drive motor will cause it to shift back and forth and it does then move it in and out multiple times while it is working at one time. The bad news is the transfer case actuator is about $1000 and the labor is about seven hours times your local labor rate. I know that's not what you wanted to hear but hopefully if you can get it to start to engage and disengage by using it multiple times it will free up. This normally happens due to lack of use probably from the prior owner. The wheel locking up in the back it's possible the rotor or caliper could be hanging up. Once it breaks loose then your normally fine. I would have the brakes inspected if you have not done so already.
The center differential lock simply engages the front and rear wheels so that you have no slip from front to back. The criteria to get the center differential lock engaged can be quite troublesome sometimes. I know it has to be in neutral and temperatures have to be a certain way. Essentially this is designed for off-road in situations where you're in mud and it's deep and you need all four wheels spinning with no speed differential or slippage front to rear. You can simply continue driving Using the four-wheel drive as normal without the center diff lock it won't hurt anything. It will perform just as good and most people never notice the difference. That picture you sent looks like the coil spring might be broke that in itself will not cause it to lockup as you have described but you need to have it addressed so that it doesn't continue to cause further problems. Should not be displaced like that if it is in fact out of the seat cushion.