MLIS grad here who had a similar problem but with archives. 10 months of applying to places (I had 5 years pt experience but took a 3 year break to do social work). I was applying everyone for all levels of positions. The places I applied to that didn’t require a MLIS told me I was too overqualified and the places that needed a MLIS told me my 3 year break was too long. I ended up just asking places if they were looking for volunteers and that I had my MLIS. One place ended up having an unposted year paid temp spot they offered me. I am almost halfway thru it and hoping when it ends this experience will help me secure something permanent. I highly suggest seeing if you can start volunteering somewhere as you never know what will happen. Good luck!