If our private insurance doesn't cover a claim, can we send it to Medicare?

You've asked several different questions about Medicare, so let's take them one at a time. First, you can submit your husband's medical bills to Medicare even if you've also submitted See also:
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See all 296 questions about Medicare, Medicaid and Medigap them to your work-related insurance. Which coverage pays first depends on the size of your employer. If you work for an employer with 20 or more employees, then that employer's group health plan is the primary payer, with that insurance paying first and Medicare paying some of the remaining costs. If you work for an employer with fewer than 20 employees, Medicare would be the primary payer for your husband, which means it pays the bills first, and the private health insurance would pick up some costs that Medicare doesn't pay. Since you've been submitting his bills to your insurance fist, it sounds like that private insurance is the primary payer in your case. If so, you can and should have your husband's doctors and other health care providers also submit their bills to Medicare, which can pay deductibles and copayments the private insurance doesn't pay. To get more details about how Medicare works in these situations, download Medicare's booklet Medicare and Other Health Benefits: Your Guide to Who Pays First.

With regard to whether Medicare requires you to keep your private insurance coverage for your husband, the answer is no. He has a right to his Medicare coverage regardless of what other insurance might be available to him. If he drops out of your private insurance coverage, Medicare will become his primary insurance.

If you drop his private insurance coverage, he can get prescription drug coverage in either of two ways through Medicare. He could enroll in a Medicare Part C Medicare Advantage plan that includes drug coverage. Medicare Advantage plans include all the coverage of Medicare Part A and B, with restrictions on the doctors and other providers that can be used. Some but not all of these plans also include drug coverage. Another, simpler option is to enroll in a Medicare Part D prescription drug plan. These plans are offered by private insurance companies and cover some of the cost of most prescription drugs.