Was reffering not if the plug will rotate in the jack, but if the wire will rotated relative to the plug, how it shouldnt br.
The wires are insulated and still work tho
It's true that the shrink tubing may make the cord more rigid where applied, depending on the tubing you use. But the trade-off is you get to continue to use your headphones.
It's sometimes easier just to replace the device, but if you really need to fix it you can buy the heatshrink tubing with glue inside (get a size big enough to fit over the connector, like 1" dia) and just kinda squeeze it together while it's burn-your-hand-hot. It's crude but works.
Alternately get a hot glue gun, put a dab in that area and quickly (while it's molten) wrap the area with some type of material like electrical tape carefully.