The solenoid you hear is actually the relays clicking in the 4x4 module to command the transfer case motor. There are contacts inside the transfer case motor to let the computer know what position the transfer case is in. If the module cannot get the motor to turn to the exact point that it wants it to be in, or if the contacts are not making a good connection, the light will flash, indicating that you are not in 4x4 mode and that the module sees a problem or cannot command the motor. That much can be a misalignment inside the transfer case shift motor among other things... Not a big deal in and of itself.
Something else you can try is to take the t-case shift motor off and manually put the case in 4hi... you will have to turn the tang that sticks into the motor with a pair of pliers and it is marked on the T-case where each position is. Start the truck and see if the 4x4 light is on. If it is, it could be as simple as the shift motor was not lined up correctly. It could be that the motor is just bad and has worn parts.
The other symptoms you are talking about are what makes me think you have a ground issue. (and it may be 2 different issues all wrapped into one).
Could you list where all the grounds are so I can check them?
Nope!
I have NO idea where all of the grounds are on ANY vehicle. The first one I would check would be the one at the battery. There should be a large and a small cable that comes off of the negative battery cable. The big one goes to the block... and if it is like the Bronco II, it stops along the way with a ground to the ECM and attaches to the frame rail. The smaller wire from the battery should go to a bolt on the core support.
Beyond that, you are going to need a shop manual. I had to buy one for my van when the dome lights kept blowing fuses for no apparent reason.
Electrical problems can be a ******* to track down. It just takes time.
The more information you give us, the better the chances are that someone can help.
Good luck!