Is it possible to begin the search and be able to discuss with a professional options, etc. by email only?

It's been my experience the worst communications have been through e-mail. That is worse than telephone communication, which is only slightly better. You really need to be physically present, or a family member or friend, or a social worker/geriatric care manager. Otherwise, you are a voice on the phone, or lines in an e-mail. (I know it CAN be done and is, but better to have someone on your mom's behalf working here in real life. Because you just don't know.
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Susanna, I'm not sure if you're thinking that this site coordinates or arranges for living facilities, or if you're wondering if e-mail would be just as effective in identifying an IL facility in the NH area as personal contact. If the former, this is a forum for posting and answering questions and sharing information. But all the legwork is up to the individual; no one here would be providing information on specific facilities.

What I would do in your situation though, since you obviously have Internet access, is to research IL facilities in NH. This does assume that once you return from overseas, you won't be returning back, or redeployed if you're in the military.

But I would first do some research on what's available and what you want. Do you want a continuing care community with higher levels of support if eventually needed? What other features are important in the facility? Individual, or part of a chain such as Sunrise? NH is a such a cold weather state that the issue of whether or not visiting physicians are available might be relevant. I wouldn't want to take someone out in a frigid NH winter, or N'Easter or blizzard.

Some facilities are like little communities, with hair salons, and I believe some even have small bank branches. Do these features appeal to you?

You could even contact the FL facility where your mother lives now and ask about recommendations in NH, prefacing your introduction carefully so they don't think your mother's unhappy where she is. OTOH, they might not be willing to offer any information if they think they'll lose a client.

That's another issue: has your mother signed a contract to be there for a given time? (I'm not familiar with IL arrangements but I assume some type of contractual arrangement is involved.)

Contact all the facilities that seem nice (of course, they all look like palaces in their online photos!),

Make a checklist, contact facilities by email, ask for specific answers to your questions. Then you can shortlist the companies to actually see in person when you return stateside.

I would look into the services of a geriatric care manager, they could do the research and present you with the available options.

If you are using the "senior living" tab on this site, I would write in the information that you've giving us, that you have unreliable phone service and would like to communicate via email.

This site is for caregiver support. Best option? Google continuing care communities in NewHampshire.

Does mom have dementia? If she is fine where she is, unless she wants to move, leave her be. If she is competent she has the legal right to live where she wants.