ABS light stays on?

It could be the wheel speed sensors, ABS module, rear speed sensors are a few other items. The first step would be to scan the vehicle and see what code(s) are in the system. Without that you could replace every part and still have the light on because it is something else. Because it's a Ford I would suggest getting a copy of FORScan and the correct dongle That will give you a tool that can do as much as the OE diagnostic system but for a much lower price, and it works on pretty much all the Ford makes and even on some Mazda vehicles.

https://forscan.org/home.html

Or get one of the handheld units that can read ABS codes.

https://www.youtube.com/watch?v=rTtAnsOlZU4

Then use the codes to give you a map of where to start looking.

This guide can help as well:

https://www.2carpros.com/articles/7-common-symptoms-of-a-bad-automotive-bearing-hub

Which tells why the bearing hub can cause this problem, but like Steve W says we need the codes to see what's going on.

https://www.2carpros.com/articles/anti-lock-system-testing-and-repair-abs

Here are the ABS system wiring diagrams so you can see how the system works. I would also check the fuses and grounds of the system which I have included as well. Check out the images (below). Please upload pictures or videos in your response to the problem so we can see what's going on.

The scan will help tell us which sensor is having an issue.

Here is a guide to help you test the sensors manually which could help:

https://www.2carpros.com/articles/abs-wheel-speed-sensor-test

Please go over this guide and get back to us.

You need to scan it to see why the ABS light is on. None of the parts you replaced would repair an ABS fault as he said. The ABS is electronic and involves wheel speed sensors, the ABS module and the associated wiring. That is why we said scan it first.

The code would clear itself if there is nothing wrong. ABS is a continuous monitor system so it is always testing, Whatever the code is should point you at the problem. If your scan tool can read the wheel speeds you can look at that as well, if it shows the rear at 45, front left at 45 but right front at 0 you know where to start looking. Similar if it showed both fronts at 45 but the rear at 0.

Should still be a history code but with the light off it may not show depending on the scan tool. Ley us know what you find.

It has an ABS relay but it's stuck behind the center of the dash (thanks Ford) which makes it a bear to get to.

The relay just powers the ABS system, won't turn the light on itself. That is done through the ABS module itself. Removing it would turn off the light because it would also turn off the entire ABS system.

Okay, that code basically means that there is a problem with the signal from the rear speed sensor. It isn't receiving the signal as a steady thing. Commonly that means a broken wire in the harness to the sensor. Try this test, go into the live data on your scan tool and select the wheel speed sensor data. Now drive on a straight line at about 30 mph or so and watch the speeds. You should see them all saying 30 mph. I suspect the rear speed will be erratic and so drop outs. I know you posted that you got a new connector for that sensor, what was the reason for that? Damage? Corrosion? Both of those could cause this code. It is also the reason the ABS light comes and goes, sometimes the sensor signal is good then it drops out. That is usually the wiring and not the sensor itself. There are 2 wires from the sensor to the ABS module. If you know how to splice the wires you can just use two wires and replace the harness from the sensor up into the harness at the rear of the truck. They fail normally because the wiring has to flex as the rear suspension moves. If I had it here, I would take the new plug you have and de-pin it and add new longer wires to it and then splice it into the harness where I circled in the image.

Yeah that would limit what you can do if you don't have a lift. I'm surprised that nobody is willing to replace a simple connector. What part of the world are you in? Maybe someone can recommend a shop.