If you have tried a new audio cable and the problem still exists then as you say the right speaker functions OK on BT then the problem is either internally in the audio socket or is between the audio socket and the circuit board in the headphones.
By convention the right headphone speaker is connected to the "ring" connector on the audio socket (see image - click on image to enlarge for better viewing)
If you have an Ohmmeter you can use it to test if the right speaker line in circuit is OK. Remove any batteries from the headphones before you commence testing with an Ohmmeter. (An Ohmmeter is incorporated into most DMM - Digital MultiMeters - these are available from most large hardware stores for around $10 - low end model but it will do the job)
Plug an audio cable into the headphone socket but do not plug it into anything else. Then connect the Ohmmeter test leads between the tip and the sleeve of the plug at the end of the audio cable. There should be a reading on the meter. This also proves that the meter has been set up correctly. Next do the same with the test leads between the ring and the sleeve of the plug. The meter will probably not respond proving that the right speaker circuit is faulty.
You will have to open the headphones and then connect the Ohmmeter between the ring of the plug and where the audio socket appears to determine now if the problem is in the socket itself (ring connector which has lost tension) by testing for continuity through the socket or further into the circuit. Do it first on the tip as you know this works and it will help you to determine where the ring connection should be.
If you prove there is continuity of the connection through the socket then you will have to look at the connection between the socket and the circuit board. If the socket is directly mounted on the circuit board check for dry solder joints around the socket area (look underneath the board if it is the "through hole" type of board) or even a loose socket (if it is the surface mount type). If the socket is "wired" separately to the circuit board check the connections on the socket and on the circuit board.
If you have determined that the problem is not in the headphone socket and you cannot see where else it may be post some close up pictures of the circuit board near the socket (both sides) and someone here no doubt will help you.
Adding images to an existing question
Hopefully this is of some help.