I am going in "I" brancher so I know more about that but from what I've seen CTs in general can get DIRSUP orders which is technically shore duty but you can go out and assists on/with ships, subs, and squadrons. As a CTI I had to sign a paper saying that I'd volunteer for Aircrew and sub duty but since I am a female only the aircrew applies to me. I am not 100% sure so take it with a grain of salt but I've seen the same info for CTRs and CTTs so I'd assume its the same for CTN as well.
https://www.youtube.com/user/NavyGypsy Im pretty sure she said she was a CTN in one of her earlier videos so maybe she can give you details!