Four way lights stay on with engine off, start and stop car and they go off

The flasher unit for the four ways is separate from the turn signal.
If the rear light is flashing fast means the front light is malfunctioning. Check all wiring from the dash to the light itself. I had one a wire was pinched and had broken

check battery, also alternator

could just be a faulty relay, if you have not had a problem before i would think it was the relay.

Have you checked the fuse under the drivers side dash?Manual should give you an image and should be label'd

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It is all related and the connections should be undone immediately. I am worried about you having an electric signal going into your gas tank which the fuel gauge gets its readings from. The signals are suppose to work on grounding and low resistence, not strong voltage.

Your Ignition switch needs a signal from a switch on the brake pedal lever which tells the truck that your foot is on the brake. There is also a switch under the dash which can be bumped or broken by hitting it with your feet. The loss of signal is suppose to keep the Vehicle in Park.

Emergency flashers work through the MFS multifunction switch on the turnshift handle. Separate flashers create the pulse to flash the lights. Improper wiring could burn out the flasher units.

On Later Ford products, the plastic cover on the Light hookup had the wiring terminals all identified. If you used an aftermarket kit, the color codes should have been assembled to match the factory colors. There is no guarantee the aftermarket units did do that.

Sometimes working on wiring is difficult if the mechanic is colorblind. Some of the better kits will have a wire connector to install in a plug under the dash between the brake pedal and ashtray. Then this harness runs as 1 piece all the way to the rear bumper. The short wire kits tap into the rear light sockets.

Some of the Electronic light flashers can sense how many bulbs are in the circuit to maintain a steady flashing rate with or without a trailer hooked up. Thats about as much as I can help you with.

sounds like the flasher in the fuse box you may have 2 //1 for blinker the other for 4 way flasher . turn on the 4 way flasher you will hear it clicking under the dash find it pull it out and find the other one that looks the same and swap them they should be sliver and round in the fuse box

There is a Rear end disconnection of the Wiring harness, most likely in the Trunk.
Regards, David

chack the grounds going to the front lights,

The most common cause of your problem is the Flasher Unit. Functioning four way flashers doesn't mean the existing unit is working. Also, there are multiple flasher units on some cars. Since the bulbs are all working, replacing the flasher unit makes the most sense.

Sounds like an earth / ground wiring problem at the rear lighting units, when the flashers are on and you press the brake pedal the power is back earthing through the flashers. Check all earth/ground points at the rear lights.....Good luck hope this helps