Check engine light?

It sounds like your having a problem with the transmission going into reverse. Let run the codes to see what comes up. Here is a guide to help you.

https://www.2carpros.com/articles/checking-a-service-engine-soon-or-check-engine-light-on-or-flashing

Please run down this guide and report back.

If the popping noise is coming from intake manifold.
You will need to check the engine timing.

Yes, because you are having the problem with the rear wheels turning. Call around to some salvage yards and see how much for a used one. Good luck let me know what happens.

Have you checked your fuel pressure? It sounds like your regulator is bad. Please advise.

Fuel pressure
5.4 & 4.6 engine;
key on (engine not running) 35-45 psi
engine running 28-45

6.8 engine
Key on 40-50
running 55-65

CEL/SES On

What you need to do here is have the computer scanned for OBD2 codes, there's a problem within the engine management system that caused the light to turn On, this is your starting point of diagnosis

If the light is on it you have a problem and need to get it scanned for whatever code(s) are stored in the system. There are codes or the light won't be on. That is the first step. It could be an EVAP code, converter code or virtually anything, many codes cause no driving issues but are still there. Get it scanned and post the code(s) and we have a starting point.

As for removing the bulb, that won't work. Neither will just erasing the code to turn the light off. Here's why, During the test the first thing you do is turn on the key and verify the bulb comes on during the test. No light on = no pass. Next if the tester says "Well maybe the bulb did just burn out". They connect up to the van. The test will cycle the lamp on and off and ask if it is working. Tester punches Yes, thinking a bad bulb. The next thing the test does is look at all of the various system monitors that are set as the van is run and driven in normal use, in most states you are allowed 1 incomplete monitor for that year vehicle. These are reset as soon as you cleared the codes, and depending on what the code is the tests may never run with the code set and the light on. Any more than one monitor not complete = Fail. The last thing the inspection machine does if it has made it this far it to scan for any current codes or pending codes that can turn on the light as well as if the light is currently being commanded on by any code. If it finds a code or lamp request = Fail. Doesn't matter if the bulb is there or not as the PCM still shows all of the above to the inspection machine.

Whoever was telling you "Just drive it, it's nothing to worry about" wouldn't be anyone I would ever take a vehicle to for anything. The reason being is that there is only that single warning light. So let's say the problem is an EVAP code for a small leak. It won't cause any driving problems but it turns the light on. It could be as simple as a bad gas cap or a sticking vent solenoid. Both are not real expensive to repair. However as you are driving another code trips and calls for the light to turn on. With the light already on you cannot tell there is an issue until it is too late.
Sorry to sound preachy but I cannot stand shops that won't actually diagnose an issue on a vehicle and give the owner the truth.

Have it scanned and get the actual code(s) that will give you a place to start. Many of the chain stores will scan them for free. Just post the codes, Ignore what they tell you to buy as repair parts, LOL. We can go from there. I've driven a lot of vehicles where the light is on but nothing seems wrong. Thing is, the light rarely comes on for nothing. In your case it sounds like it could be a simple wiring connection. The codes should help.

Okay, to narrow it down E150, 250 or? 4.6, 5.4 or? That sensor isn't hard to test and depending on which way it is reading it could help the fuel mileage. If it's the one on the rail it's real easy to change as well, a vacuum line, wire connector and 2 bolts.