http://www.arcgames.com/en/forums/startrekonline/#/discussion/1207878/player-starship-invisible-unplayable-in-sunrise-after-11-19-patch Having the same issue and the solution given of swapping ships worked for me.
I'm having a similar issue. I can interact with them eventually I just need to find the one small spot that actually gives me the option. Either way it's a pain to have to spend 5 minutes finding the one sweet spot to interact with a console to continue a mission.