I am here in the U.S. and my visa has expired. My spouse is a U.S. citizen or U.S. resident. Can I get my U.S. residency here, or do I need to leave the country and apply overseas?

You do not have to leave the country.  You can make both a Family petition (I-130), and Residency application (I-485), in the U.S., and stay in the U.S. during this immigration process.