The only way I've found to test if the fronts actually engage fully in 4Hi or 4Lo is to strap the Explorer to another vehicle on dirt. Engage 4Lo and put it in first gear. Apply power until the tires spin and see which are spinning.
I can also flip my Brown wire button on a wet road, prove that I can spin the rears, than flip the Brown wire button back off so all functions normally and see if I can still spin. If not the system is working. This is a little more difficult because you can fool yourself since its very hard from inside to tell if there is any front spin or not.
Another possibility is engage 4WD (any but Auto) on a dry high traction parking lot and turn the front wheels hard to either direction and drive in a circle. The drivetrain should bind but this is more difficult to know for sure that 4WD is working.
My final way to positivly know if the front axle is engaging I would not advise anyone to do but I have. Rear wheels up on jack stands, fronts on ground, left foor firmly on brake, engine running. Now shift from park to nuetral and engage any 4WD mode, even leave it in auto. Now shift to drive and slowly let up, but not off of, the brake. The fronts will try to pull you off the jack stands if the TC is fully engaging. Actually, if you let off the brake much the fronts will pull you off the jack stands and you may damage the vehicle, bystanders or yourself.
I prefer the method of strapping the Explorer to another truck. Its fool proof, assuming Im the fool. I have trouble shot this transfer case many times, the clutch pack itself seldom fails but it can.
The easiest place to access the Brown wire is under the drivers seat. It is easily seperated from the bundle here. If the seat is removed, you can even follow it down through the floor where it exits beneath the vehicle by itself and enters the transfer case. 12 volts applied to the brown wire at this point will fully engage the electromagnetic clutch (a ball ramp design) and lock the front and rear TC outputs together. This works much like an electronically locking differential.
There are two other aspects of this TC I see frequently mentioned but I have never been able to verify in or out of the truck. 1) The clutches always drag a little, even when disengaged. If so, no more than a loosly set-up Limited Slip differential. There is simple too much space in the ball ramp itself and the clutches are wet. Im not saying none but really almost none. 2) The transfer case is capable of supplying varying amount of torque to the front output. That is partially applying the clutch. First, this is a ball ramp design. There is no way to hold the balls partially rolled out. Even if the controller was PWM (pilse width modulated) this wouldn't work. The signal sent down the brown wire is either all or nothing, not a varying voltage. So it is off or on. Albeit, it can and does pulse the clutch. Just not fast enough to partially engage the clutch and send a varying amount of torque to the front axle.
P.S. Once the transfer case clutch is activated, it takes about 30 degrees of rotation to fully engage. How much wheel spin, in Auto mode, for the electronics to decide to engage the clutch? Well that varies a little as the computer gets info from a variety of sensors and compares them. Generally less than one rotation of the rear tire but some times slightly over one rotation. How long does it stay engaged before unlocking and looking again to see if there is still wheel spin? Not alot but it seems to be a little hard to pin down. It seems to be about 4 rotations. My tests have had too much variability to be acurate.