Is it a good idea to connect ethernet cable's shielding to PCB GND?
'Regular' Ethernet cable (Cat3, 5, 6) is unshielded. That's the U in 'UTP' - Unshielded Twisted Pair.

As for the connector, RJ45 as used for Ethernet provides no grounding at all. This is intentional: the Ethernet magnetics actually completely isolate any DC path between the cable and the electronics, so there is no common-mode path at all, ground or otherwise. This is fine for Ethernet which uses an AC signal, but for an ADC input that presumably needs DC, this would pose a problem.

The shielding you see on the RJ45 connector doesn't actually make a connection to the cable in any way. Instead, its purpose is to suppress common-mode radiation from the board itself and prevent it from coupling onto the transformers (if it's a magjack) and cable pairs.

For your application, HDMI cabling would perform better and provide the common-mode shielding you're looking for. HDMI cable provides 4 shielded pairs with separate drain wires for each shield, plus an overall shield that connects to the shell. You could adapt this to the connector of your choice (not RJ45) or use an HDMI connector if you wish.