That is an important concern. The best bet would be for you both to purchase long-term care insurance, but that can be expensive and there is no guarantee that both of you are insurable.
The next best step is to sign a prenuptial agreement that covers long-term obligations and subsequently to keep your finances separate.
Depending on the state you are in, should you need nursing home care, your new husband should be able to refuse to cooperate with your application for Medicaid coverage. This approach is often called “just say no” or “spousal refusal.” I would recommend consulting with an elder law attorney in your state to see how this strategy works there. Find an attorney in your state.
Also, you may want to check out this article on the pros and cons of remarriage for older couples.