Four wheel driving not working properly

I have a 1998 Ford Explorer with on demand 4WD. I have had the vehicle looked at by Ford repair shops as well as independent shops and have had numerous fixes applied costing me lots of money but the problem remains unresolved. When starting down a mountain road right after turning the car on I engage the 4WD low in 1. The vehicle starts to walk down the road but only gets 1/4 mile down before the 4WD lets go and I find myself freewheeling down the road. I have to apply brakes from that point on.
All the shops engage the 4WD on flat surface or on a lift and see the four wheels turning and tell me they have fixed the problem. Since there are no significant hills where I live I cannot demonstrate the failure to them. Any ideas on what to point them to so they can fix this right

It is electronic. There is a switch on the console that allows you to go from 2WD to 4WD high and 4WD low. There is a transfer case motor that engages when you move the knob to 4WD.

Thanks for getting back to me. I believe that even when the 4WD cuts loose the 4WD Lo light stays on. Is that helpful? What do I tell the mechanic to look for since they all believe "there is nothing wrong with your 4WD" I believe your explanation makes a whole lot more sense. Look forward to your answer

Thanks for taking the time to read over my car problem. I can tell you that what you've said makes a lot of sense compared to the other advice/troubleshooting I got from the different shops. A Ford dealership put in a new transcase motor a few years ago and I suspect the work was very shoddy. The reason they did it was because the console light was flashing between normal, 4WD Hi, and 4WD lo while we were driving on the highway. Hopefully I can find a shop who can do the work.