Regarding getting away from Mistyra, it's possible your stats are too low. She likes throwing the meat around, and your stats need to be at a certain level to be able to resist/defeat her. There's also (I think) a move that takes a turn to make work. Like if the female parasite says "She turns towards you", DEFEND to avoid the attack.
Regarding getting rid of the parasites, the Science lab in New Ark is your answer. They'll get rid of the parasites for you! Warning though, next time you visit you will need your stats somewhat high (strength nimble, and resistance should all be over 12, you can bulk up at the gym) or you will find it VERY hard.