Re: 03 Ford F250 6.0L 4x4 schematic.
If you have the electronic transfer case there's not much you can do except to replace the TCCM, or transfer case control module. Check for used first, as new is not only twice the price, but a used one can have similar reliability. The truck is 7+ years old.
I'm not sure what relay you refer too under the hood, but if you were to replace that relay even you might be OK and the TCCM might be fine, as the TCCM is what controls the movement of the transfer case forks when you turn the knob to a different mode.
Even with a manual lever or electronic knob on the dash; if you allow the truck to roll back slightly, it can help disengage the current mode if the axles are bound up a bit. In your case this more than likely won't be a solution if a 'relay' is acting up...
'11 Ram Quad Cab 4x4 SXT 5.7L HEMI (stock); 48,500m
'06 Pontiac G6 GT 3.5L 220hp; 64,000m
'12 Chrysler 200 Limited 3.6L 283hp; 28,000m
'99 Taurus 3.0L 2V Vulcan 145hp; 154,300m - Traded
Amsoil in all vehicles!