The first concern that needs to be resolved is whether your state or insurance company recognizes domestic partnerships as legally binding in the way that a regular marriage would be. If your state does not recognize domestic partnerships but your insurance carrier does, you will still be covered. If neither the state nor your insurance company recognizes same sex domestic partnerships, your options are somewhat more limited. Some insurance companies recognize domestic partnerships even in states that do not, so check with the insurance company to determine how the company handles such situations.
If your insurance company does recognize domestic partnerships, you should qualify for the COBRA health insurance extension. COBRA was passed into law specifically to prevent people from losing their health insurance immediately on job termination or some other circumstance that could impact your insurance. Under COBRA, you will have to pay the full cost of your premiums, but your coverage will not be interrupted until you have had a chance to find replacement coverage.
Another option you may not be aware of is the ability to split the health insurance contract, effectively turning one health insurance plan into a two separate plans. Both parties would end up with their own plans, but the coverage would remain exactly the same as it was before your split. Be aware, too, that depending on your state laws, either you or your partner may be required by the court to pay for the coverage for both parties for a specified period of time.
If all else fails, you may qualify for government sponsored health insurance through the Medicaid plan. You will have to meet income and/or have a preexisting condition that makes you uninsurable under normal insurance, but Medicaid will allow you to have insurance if you cannot afford to purchase an individual or group policy.