So I'm fighting the Borg during the Salvaged Relationships in the Neutral Zone (which is its own problem altogether) and I die. I respawn, but my ship turns very very slowly and I can't use Full Impulse. So I leave to sector space. I can turn again, I think "great...problem solved." I go to K7, and again...can't turn and no full impulse. I go to Sol. Same thing. I exit the game and restart it. Same thing. The game is basically unplayable. Guess I'll reboot and see if that helps.
So it turn out I was missing my Impulse Engines. I had picked up some Impulse MK II while fighting the Borg...switched them in and sold my MK I then died. I guess I subsequently lost my MK II and boom screwed.