If the connector plugs into wires coming from the brake pads it's for a brake wear warning light. ABS sensors are in the swivel, aimed at the CV.
There'd also be an ABS block sat under the bonnet, if you have ABS.
I'm guessing the handbrake is on, which works on the back prop and axle ....
1/8 of a turn sounds normal. play can come from any of those areas, though remember wear after the diff is going to feel like its 3.54 times worse if you're feeling it through the prop.