Nice to see a few balanced posts here.
OP, the things you need to make sure of are:
1 - Health insurance that has a very low annual deductible for individuals and family as a whole, and that has a low or no co-pay, and a high payout rate (over 80%) with a low maximum out of pocket expense limit.
Be sure you understand the terms used in health insurance policies:
https://international.iupui.edu/indianapolis/health-insurance/glossary.html
2 - Whatever city you end up in. look carefully at schools. Schools are organised by District, and boundaries are strictly enforced in most cases. In general, better off areas have better schools (due to the way public schools are funded). Research schools carefully.
Public schools are obliged to accept all students residing within their boundaries, so you won't end up with the sort of lottery system operating in many areas of the UK.
You will be required to show that you live within the boundary of the District, however. Registering a child for school usually requires proof of residency (your annual lease agreement if renting, driver's license showing picture ID and address, mortgage documents if applicable, mail from a bank / CC issuer showing your name and address, or mail from (for example) the Secretary of State's office wrt driver's license - official mail at any rate, and child's birth cert). There is sometimes a reasonable/ small registration fee. Be prepared for a minor kerfuffle when you present a foreign birth cert 
Private schools don't require proof of catchment residency. They will need a birth cert, and you will pay tuition. Private schools can be independent/non sectarian (like Montessori), or can be run by organised religions (RC, Lutheran, Presbyterian, Episcopalian, Jewish). Many offer wrap around care, which public schools don't often do.
Preschools are often found in private schools, but you can also find preschool offerings run by local parks and rec departments, YMCAs, and community organisations.
3 - Your children will need to have Tennessee-mandated vaccinations done prior to attending school. You will need to find a pediatrician strictly for the children or family practice doctor (who would see the whole family) as a matter of priority. Children do annual well-child exams in the US in most states. Children are also often required to do an annual dental exam and present proof of this to school. You can look up whether TN has these requirements.
Since your DH works in an industry of a "secure" nature, I suspect you are not necessarily looking at Nashville or Memphis (though I may be wrong). There are nice places to live in every state; don't be put off by smaller towns as long as you're not looking at a very run-down, rural dot-on-the-map deep in the mountains.
I've never seen anyone carrying a gun in the US apart from police officers. My DCs have all done tornado and fire drills in school, as well as lockdown drills. They've taken it all in stride. We've done fire and tornado planning at home too as we are in an apartment.
The cost of living will definitely not swallow up most of any pay increase your husband gets. This is the weekly ad of a pricier/fancier supermarket in the SW Nashville burbs:
https://www.publix.com/savings/weekly-ad?aid=1_hp_mg_mb_weeklyad_savings
Just as you would in the UK, you can take advantage of sales, BOGOs, and use your freezer.