Since your husband has Medicare because he's over 65, Medicare is the primary payer of his covered medical bills, with COBRA coverage through your work being a secondary payer. You Medicare and Other Health Benefits: Your Guide to Who Pays First . If you want to go deeper into the technical rules, you can find them in the Medicare Secondary Payer (MSP) manual .
can find this rule explained in the Medicare booklet
You and your husband may also want to consider whether he wants to continue with COBRA, or whether a better idea for him is to buy a Medigap supplemental insurance policy. In making this decision, he has to consider that the open enrollment period for Medigap, during which he can buy any policy offered in your state regardless of his health history, runs for only six months after he enrolls in Medicare Part B. To get help with this decision, he may want to consult, for free, with one of the expert counselors at a local office of the non-profit State Health Insurance Assistance Program (SHIP).