From the description of the symptoms, you get the impression that you were using the volume knob, not the balance knob

It's so off the beaten track.
I would suggest two things:
1 - checking the headphones on other equipment that has the ability to adjust the balance
2 - Hope you have access to a meter (digital multimeter). I am writing about it because by measuring the resistance seen from the plug, you can determine whether the connections are correct or locate any irregularities.
Assuming you have access to such a meter, take the resistance (or resistance) measurements as follows:
a - you need to know the resistance of the handset - if it is not known, unsolder one of the cables by the handset and measure the resistance in the place where you are soldering the wires. After the measurement, solder the wire back on.
b - take measurements on the minijack plug. Note here - take measurements on the plug itself, not on the side where you soldered. For measurements, the plug should be folded, i.e. as in everyday use.
1 - plug tip and ground - correct reading -> value that you measured on the handset
2 - middle section of the plug and ground - as above
3 - tip and middle section - correct reading -> doubled value of the handset resistance.
If you enter the results, we'll see what's next.