suggestion: make many copies of the letter from HR
call your local labor department and tell them the situation and ask for their advice and/or file a compliant
Many employers only give dates of employment, pay, and title now anyways for references.
Look into NAIS at private/boarding schools. Even one that is not in your state because you could get on campus housing for you and your children to live in.
Try to take a step back and don't be so hard on yourself. I know this is tough but don't panic or think the worst.
I also do not think you should resign, unemployment lasts for a long time now with extensions, and it may take a while to get a job.