My guess is that your priorities should be in this order:
- Get coverage (almost any coverage) in place for you and your children
- Protect your position vis-a-vis the other parent by communicating clearly what actions you are taking and why
- Explore opportunities to reduce costs, like obtaining employment with an employer who provides group health insurance, shifting insurance to the other parent, and purchasing insurance on the private market or from your state.
- Explore any rights you have against your former employer
My guess is that your former employer IS covered by COBRA unless your work force is smaller than 20 employees.