DON'T quit your day job!!!
Demand is not that high. If you have benefits, stay where you are. If a good part-time job opens up, consider it, but these days it may not lead to full time - municipalities are under the gun to reduce full time positions nationwide. I know someone with excellent skills who moved to a preferred location and took 5 years to find a full-time job.
Use your current position to build contacts in the public library field. Public librarians are usually EAGER to have an ally in the schools who can alert them to upcoming assignments. Invite them to your library and visit theirs regularly, even on your own time if necessary. Stop in to check out a few books and say hi. Ask what the kids are reading; compare notes. Volunteer on your own time, directly or through a Friends of the Library group. Be seen, be pleasant, schmooze - with sincerity.
Academic Libraries are tougher, unless you are with a high school. In that case, you can visit and ask them what you can do for your students to help prepare them for college. Ask if a librarian would come and talk to the students about using the college library. If you are in elementary school, and they have offer an Education major, make available a little video tour of your library online (sans kids, of course, and assuming you are allowed) emphasizing the value to future teachers of these great resources.
Cataloging and research are not really widely available library jobs anymore. Reference, yes, but not much detailed research, except in Corporate. CIP and OCLC obviate the need for most in-house catalogers. Maybe a central branch of a large public library system would have a few; I'm not sure about colleges. If you don't like facing the public, it will take a long time to find a good fit.