In some states, you can maintain the coverage for an extended period -- often as long as another 18 months. If you still haven't found another job that provides health insurance after that, you should contact your state's insurance department to discuss your options. If you have maintained coverage for the last 18 months and have never let it lapse for more than 63 days, you are entitled to buy health coverage that has no limitations on preexisting conditions.
I have a child with diabetes. When my COBRA eligibility expires, is my only option to sign up for the state's high-risk pool?
No. You should actually consider splitting your family coverage. If everyone else is healthy, you may be able to get a low-cost policy through the individual health insurance market for everyone else. That one child could stay on group coverage through COBRA, or get coverage through the State Children's Health Insurance Program.