Shield stays connected to the chassis only. Ground is pin 1. I use Neutrik's that have separate shield/chassis connection from ground and chassis goes back to the PSU where only there chassis connects to circuit ground and mains ground (in a star manner).
So, in your example for the input I would only switch pin 2 and 3 (and not switch ground, nor shield) with a DPDT relay best, something like a a Kemet EC2-12NU (one per channel), located near the inputs on the back of the amp. A front located SPDT switch will only toggle the relays.
The output I probably would not switch at all, but depending on the subsequent device (power on/off status) it will be the one or the another (or both, if you make sure you can use both outputs at the same time). Ofc, you can have the same switching for the output as on the inputs, one EC2-12NU per channel and switched by a front located simple SPDT switch. As you prefer.
The above Kemet's are for 12V, check the voltage in your device or use MC78/79 type of voltage regulators or change relays to 5V,... 24V to suit your needs.
Omron G6A-234P-ST-US relays might be used as well.