The 2 front blinkers work. The odd thing is when you push the turn signal on, whether right or left it turns on the opposite side you turned on. So I turn on what is supposed to be the right-hand blinker and on the dash the left blinker indicator lights up and the left blinker in the front are blinking. So the 2 front blinkers work just they are working in the opposite way they are supposed to.
the rear blinkers just don’t work at all. All the other lights in the rear work such as the brake lights and running lights but when you turn on the blinkers again the front work in the opposite way but the rear ones don't do anything as far as blinking. I have the
flasher our and it sounds as it is working but I ordered a new one to be safe.
Ok, that's different than I pictured it and different than anything I've ever encountered. I would not expect a new
flasher can to help this as it interrupts the circuit to the turn signal switch not the power from the switch to the right or left side. The way the switch is typically constructed an intact swtich cannot possibly cross right to left to left to right...
So what follows is an educated guess.
I am presuming as a '63 it does not have an emergency 4 way
flasher as I don't think those showed up until 65 or 66. (I had to add an aftermarket 4 way to my '63 Mercury Comet since it is a daily driver).
No 4 way removes a level of difficulty on what can go wrong. I'm coming back to the turn signal switch or worn wiring in the steering column. Although now it almost sounds like someone is pulling a practical joke on you and got under your dash and swapped some connectors around at the steering column.
The wiring activating the opposite dash indicator and front signal means the cross up for that issue is happening at the switch or column wiring. The fact that it is flashing indicates power is coming from the right place, getting to the switch, and the
flasher is good, and the front signal and dash indicator apparently have a proper ground. Power is not getting to the rear lights from the switch OR the rear lights have no ground connection is the next point to address, BUT you indicate the brake lights work. Even when a signal is flashing up front, the brake lights work on both sides in the back, right? The turn signal switch SHOULD interrupt the power to the brake lights from the brake switch on each side so the
flasher can supply power and blink the lights. SO the turn signal switch is most likely your issue. But this a failure mode I have never seen before.
A longshot, but possible, is that you somehow lost ground to all the taillights, but if the brake lights work properly that rules against that.
There was no recent work under the dash or in the trunk?
Where are you located? maybe a member can recommend an electrically savvy shop that can take a look?