I am a public librarian & paid by the city, why aren't federal income taxes taken out of my paychecks?

I had a rude awakening when doing my taxes this year. I know that the library is exempt because it is a non-profit organization. But, wouldn't they still need to take fed. taxes out for their employees? How should I make up for this difference this year? Should I have my husband raise his exemptions to make up for it? Any advice? Thanks so much.