I have the Flaw that I never break my word. My party got a contract and promised to clear out a temple. Most, if not all, of the party has run off to join the people in the temple but I am sorta trapped in having to now face my party because I have to keep my word. I also have the ideal of respecting and obeying the people above me. The contract is permitted by the Govt. as well. I feel that to play my character I don't have much of a choice but to follow through against my party. I'm not really sure what to do.

You don't always have to follow your flaws, they're just guidelines that you normally follow. You can change, but I would recommend sticking by it this specific time. It can be fun to roleplay as a character that has to stay out of a mission, or not help their party because of a strict moral code.

You always have a choice. And just because you break your word, you could reflect that in the repercussions your character feels about what they had to do and that could shape aspects of your personality going forward.

If anything, this is a good opportunity to role play justifying it to yourself and still being bothered by it afterward.