Had a neutral tow kit installed on our 2002 Ford Exploder (that's what we call it - not a typo - remember those exploding tires back then?).
Haven't had that issue of not engaging, although you have towed yours many miles more. Other than some clicking along with the lights flashing, we don't have any sounds, as you indicated yours does.
Since you are able to go manually through the 4x4 settings, I would think that it would be likely an issue with the neutral tow wiring rather than the transmission itself. You may want to check the ground. If not that, a call to a Ford service center would be your best bet.
When you solve it, would appreciate if you'd post what the issue was and the fix.
G-half can always find a way to do things upside-down, inside-out or backward.
It's his Super Power!