My husband is in his 90's and showing signs of mental issues. But he hasn't been diagnosed with Dementia or anything like that. He's been in the nursing home for about three years. No major health problems so far, other than having to wear a urostomy pouch. He can't walk very well either. So he uses a wheelchair.
Why does your husband want to move? Sometimes people with early dementia express a general unhappiness with their living arrangements that is part of the disease and no move would actually make them happier.
If his nursing home stay is being paid for by Medicaid you may have less options but you certainly can ask the new facility if they have an open Medicaid bed and if they would accept him. If they say yes, they can guide you through the procedure. The first thing they will do is assess him to see if "they'll be able to meet his needs". I think that's code for "How much care will he require from us?"
In my opinion, it's hard on elderly people to be moved. But if you think he would be happier in another facility, you can start by touring them.
Check menus, if residents have a choice of alternate foods, if there's a chef on site. The best (at one time) facility I found offered menus as well as a la carte choices for patients. The worst was affiliated with a hospital and served stale, inedible food. Seriously. I thought the fish was plaster from a wall. It looked and felt like plaster and was completely inedible.
Speak with your husband as to what he wants, i.e., activities, library access, music, pet visits or resident pets, etc., and ask each facility when they have them. Sample daily, weekly and monthly schedules are also helpful.
Let him review the materials (including the typically glitzy brochures) so he can feel that the choice is primarily his (if you think that's important to him).
Ask him to list what he likes and dislikes about the current facility, especially why he's ready to move (he could be just tired of the same place). A checklist of his preferences could help both of you decide on a good move.
Another factor would be future care, what other facilities might exist in the complex. After years of caregiving, I found a community with step up facilities: rehab, AL, IL and Palliative Care then Hospice. When my father was too weak to rehab, he stayed right in the same facility but segued into Palliative Care. Moving would have been unpleasant and probably unsafe, given his condition.