If you know which side of the stud bay the 14/2 is on, you can run the CAT 6 on the other side of the bay. That will provide about 14" of separation. When the CAT 6 has to cross to be near the 14/2, it will be crossing, not running parallel and shouldn't pick up too much interference.
UTP cable (Unshielded Twisted Pair) is designed so the twists in the cables resist cross-talk between the pairs, but also to reject external interference. Running parallel to 120v main power could overcome the cable's designed in resistance, but crossing the cable, as described above, shouldn't be a significant issue.
I believe that shielded CAT 6 cable has to be terminated with a shielded RJ45 jack that has metal on the outside which is connected to the shielding in the cable and is then grounded by the router. I'm not certain if the router has to be specifically designed to handle shielded cable, but I'd say there's a high probability. I just looked at a residential grade router I've got sitting around - some of the jacks have a metal strip on the inside, some are all plastic. The metal strip might be grounded. You'd have to look at the documentation for your router to know for sure.