On a 47 year old trailer it is hard to tell what the problem is with your running lights. It could be anywhere from the TV connector to the 7 wire connector to the wiring itself.
Here is how I would attack it: I would take one of the new tail lights off and identify which wire is the one going to the tail lights (the smaller, dimmer filament). Then I would hook the + of a 12 volt
source (battery or older battery charger which delivers 12 volts all the time, not one of the new smart electronic ones) to that wire, still connected to the original trailer wiring. Hook the negative line to the chasis metal somewhere. That tail light should light (assuming it is grounded).
The 12 volts should backfeed the other lights, that is the other tail light and the marker lights. If it does light up the other lights (or most of them, sometimes bulbs on the marker lights get corroded and don't all light) then you know the wiring to the lights is OK. I am going to assume that some or most of the lights work with this test. If none do, try the other tail light the same way.
Assuming you get the lights, or most of them lighted, take a meter and go to the 7 wire connector. Attach one lead to the metal tongue, and use the other lead as a probe to see if you can find 12 volts at one pin of the 7 wire connector. (oh, disconnect the trailer battery and unplug it from 120 volts so the converter is off to do this test, as one other wire could have 12 volts on it, the charge wire).
If you have +12 volts on one of the 7 way plug wires with the running lights on, you have determined that the trailer wiring is at least working.
If not, then I would look at the 7 wire plug connections first, clean the contacts and so on. Maybe replace the 7 wire plug, they get old and tired in a few years. If you still have no power at the plug, you will need to trace back to where the large 7 wire cable goes into the trailer and connects to the wiring put in there. On your vintage trailer, there may be a small inspection plate in the bellypan near the tongue frame in the front that gives you access to these connections. Repeat your connection inspection there.
When you get the +12 volts at one of the pins of the 7 wire connector, you know the trailer is ok. Then it is time to attack the tow vehicle in the same way. Is there +12 volt power on one of the pins of the tow vehicle, power that goes on and off with the tail lights.
You can attack this problem from either end, working back from the tow vehicle running light power to the trailer or forward from the trailer to the tow vehicle. It is hard to predict which will give you faster and better results.
Another thought, if you had someone work on the 7 wire connector recently, they should have "fixed" the running lights while they were at the other issues. How come they didn't? In re reading your original statement, you said you had a "new 7 way put in". Old Airstreams used a different color coding than is standard today. Possibly the shop which did the work has connected the wires wrong in the 7 way connector. Some shops are not too swift or don't really care. Is that a possibility?
There are so many things that could cause the problem that it is just best to start at one place and work to the other end with a 12 volt
power supply and a test light or meter. It is not rocket science, but just plug and chug legwork and understanding of simple electrical circuits.