Connecting the Shield of a strain gage cable connector to PCB with ADC
As a general guideline, it is preferable to connect any cable shields to the metal chassis (not PCB ground), at just one end. To keep things simple when multiple shielded cables connect out from some central device, the shield connection should be done only at the "hub" device, and left open at the "spoke" devices. 

Similarly, for a chain of devices connected by shielded cable, each shield ought to be connected to the chassis of the upstream device, and left open at the downstream end. Yes, this does mean that the shields of different links of such a chain might be at different potentials, depending on how well earthed the individual devices are, but this is generally not a problem. 

A basic, inexpensive addition incorporated in many consumer device cables today, is a clamp-on ferrite bead or "split bead" at each end of the cable, close to the connector. This reduces high frequency EMI off the shield with minimal complexity. A good document about such RFI beads is here.

Note that the chassis is usually connected to "earth" of the location, not to ground of your circuit board.

Of course, this is a simplification of a fairly complex subject, but it serves the purpose for designs at frequencies where surface effects do not predominate. As frequency of signals involved rises into GHz, other factors need addressing.