it looks that the cable is broken near the connector - if is cuts in and out when you move the cable near the connector, not the plug itself.

If you intend to replace the cable anyway, I would cut the cable say 10cm from the plug, to get access to the inner cables. there should be 3 or 4 signal paths: Shield (optional) equal to ground, ground (usually black), right channel (usually red) and left channel (Mostly white).

you then can strip the wires back, connect the ground by twisting the wires together, and try the headphones by touching the others together.

In your case you say get right all the time, so use the red from the plug with the left channel from the headphones and see it it works reliably, then use the right channel from the plug with the right channel from the headphones just to confirm that works too.

If it checks out, your headphones are proven ok, and you can either replace the cable, or just solder on a new plug, as the break seems to be near the old plug in the cut off portion (this is the case most often)