Well, I got a chance to fix the tail lights. I got new sockets and bulbs and grounded them along the same wire to a solid ground in the trunk. As soon as I reconnected the battery negative terminal, all the tail lights went on. Mind you, the key is out of the ignition and the headlight switch is off. When I try the directional switch (on either side), both lights on the desired side (the turning side) go out.
When I put the headlights on, the tail lights get brighter. And, with the headlights on, the turn signals on one side do not go out and on the other side, the inner tail light goes out. It's odd, because that's the one that had different wiring in the socket. The two left and the outer right tail lights all had a black and a white wire. The inner right had a brownish black wire and a double green wire. The socket was also a black plastic instead of a white plastic.