How difficult is it to utilize a nursing home in a different state from one’s residence?

An eldercare attorney is a great idea, but in fact, I might START by shopping for NH's in your desired state.

We placed mom, a lifelong NY resident in AL, then MC then NH in the bordering state of Connecticut. We did this in part so as to have her closer to one of us, but also bevause of some good initial advice from a "senior placement advisor" who pointed out that costs were less in CT and the regs were different. At the time (2013) if you were a 2 person assist, you could ONLY be cared for in a NH, whereas in CT, ALs and MCs were allowed to do this.

Mom quickly progressed to needing a NH and we found one that would accept her as a Medicaid pt after 2 years of private pay.

In the end, we didn't need Medicaid, but was good to knowcthat it was an option.

I think you should consult with an elder law attorney for the state you're considering for him. Assuming you are the community spouse so have shared assets...it can get complicated and this forum is just random individuals who aren't lawyers and even if one was, that person would need a whole bunch more info in order to give you the best legal guidance that you can place confidence in. What if you got info from here that turned out to be wrong? Or illegal? There's no recourse, not to mention all the other downsides of that outcome. It will be worth investing in actual legal advice. FYI the Medicaid app has a financial "look back" period. In many states it is 5 years.

Is your husband by chance a Veteran? If so the VA might be an option. Depending where and when he served it is possible that some of his conditions would be considered “Service Connected Disability”.
If he is a Veteran contact your local Veterans Assistance Commission. They can help determine eligibility. And with recent changes you can be getting paid to care for him.

You will need to check with Medicaid in the State you want to place him. Some Medicaid is immediate, others you have to reside there for a certain amount of time. Then you have the Community Spouse thing where your assets can be split with his split being spent down and Medicaid applied for. Your monthly income is looked at and you will get part or all of it to live on.

I live a bridge away from the closest state so placing someone in that State would not be too bad. I am maybe half hour from the cluster of ALs and LTCs in that state. Are you willing to relocate to another State?