Thank you very much for your reply, Chuck Tobias! Greatly appreciated!

So before I could even get to the point of testing the ground on the tail light assembly, I did the following:

1. Opened Trunk, unfastened the snaps for the carpet covering the Passenger Side Rear Tail Light Wire Harness/Ground Wires/and Tail Light Assembly Nuts that hold it on. One of the snaps for the carpet that prevents it from being folded back is actually a plastic hooking point for cargo netting that doubles as a nut that helps hold on the entry ledge plastic protective cover, and it just unscrews by hand in the normal righty-tighty lefty-loosy way.
2. I disconnected the Nuts that hold on the Tail Light Assembly while holding the Tail Light from the outside of the car so it does not accidentally fall off suddenly and get damaged. There are 3 Nuts that hold it on but one of mine was missing, it's probably fallen somewhere where the spare tire goes in the trunk since that's the lowest point, at some time in the past when a bulb had been changed before, but two was holding it on more than plenty tight enough.
3. I did a quarter turn or so on each of the Female Light Bulb Sockets to remove them from the Tail Light Assembly so that I could set that aside and not have to hold onto it nor leaving it to put strain on the wire harness. There are 3 Sockets in total, two big, and one small.
4. The light bulb that was having trouble was very self evident that it had burned out because it actually scorched the whole inside area that the light bulb fits into (see photos to see both a good working socket and the black socket that had been scorched).
5. I removed the bulb, which was a good one because I had already tried replacing it before, and looked more closely at the socket. The metal contacts that touch the fine wires on the joint of the light bulb were Also equally scorched with black marks left behind! I found it odd that the bulb was staying lit with the normal running lights but was not flashing with the turn signal on.

So I tried this:

1. I put the keys in the ignition and turned it to the accessory setting so that it would be on and providing power but not enough to start the engine and I turned on the turn signal.

2. I touched the good bulb into the socket just enough to touch the outer parts of the start of the metal contacts, and sure enough it was blinking the light as it should be, although at a faster than normal rate, indicating that a bulb is still out (typically the reason for fast turn blinkers). I turned the ignition and turn signal back off and removed the keys.

3. I happen to have a pen style diamond knife sharpening hone with one end that is tapered to a very fine point that's used for sharpening serrations on knives. I used the point, since it was more than small enough, I used the diamond grit of the surface of the sharpener as a kind of very small and fine grit sand paper that I could target those metal contacts where they were scorched, however just for good measure I rubbed all of them until the black soot scorches were gone. This only took a few seconds really, and I was using a tiny dab of light bulb grease (like they sell in the small packets at auto stores, since I happened to have some left over nearby), to help keep it lubricated and so that I could coat the contacts with it as I was sanding gently on them.

4. I put the new bulb back into the socket, put the key in the ignition just like before to the accessory mode, and put the Right turn signal on since it was for the Right side that I was working on, and despite that socket looking pretty awful from being scorched, it still worked just fine! So after about a minute or less of the turn signal on it finally went back to the slow standard flashing rate instead of the fast mode it had been stuck on.

5. Satisfied enough with those results, and double checking that the brake was turning it on as well as it should per normal, I then turned the car ignition off and reassembled the Light Sockets back into the Tail Light Assembly, put the Nuts back on, and before jumping the gun thinking it's fixed, I tested it again, it was still working as it should. So I then put the carpet back down and reinstalled the tabs/pins/and everything else as it was before. Shut the trunk, and tested it again one final time but this time with the car engine started and it's still working fine now.

So, I probably will end up replacing that socket at some point after I can track down the right one. I took pictures of the numbers and of how it looks for the style of it, along with pictures of the bulbs used for future reference and will include them here with a link to see them all. I don't have full pictures of the entire process but if requested, maybe I'll make that happen when I go back to replace the burned out socket. Or perhaps I'll even make a detailed video later on depending on the interest of it all. For now, I hope this might help someone else in the future!

Pictures: https://postimg.cc/gallery/b1JW7mM

Check the ground on that tail light assembly.

Usually, the bulb for the running light and brake light is a two filament bulb and the turn signal bulb is single filament, as is the reverse light bulb.  Check to see if you have the correct bulb by checking the rear driver's side bulb to see if they match.  Just a thought.  You might be needing a new flasher relay, but since the driver's side seems to be working, it's probably not the case.