While you can do it on a mountain bike, it will be noticeably easier and more comfortable on a touring bike. Mountain bikes are made for tackling rough terrain. They’re not built for comfort or speed. Touring bikes are made for covering significant distances. The frame geometry, gearing, steering geometry, handlebar design, tires, etc. are all designed with the intended use in mind, and it does make a difference.
A rear rack and panniers could probably carry what you need in terms of clothing and basic gear. A handlebar pack is incredibly helpful and convenient for stuff you want within easy reach. Unless you don’t see a need for maps, a handlebar pack with a built-in map case is what I’d recommend. To carry more stuff, front panniers are available, but probably not necessary if you’re not camping and pack things efficiently.
I found a bike computer really handy. Mine had readouts for current speed, average speed, mileage covered on total trip, mileage on this leg, etc. I also liked having a rear-view mirror.
Headwinds slow you down considerably. Sidewinds are a constant fight. Tailwinds are a blessing. Bear in mind that the wind can shift for any number of reasons, and you can’t depend on their always being favorable, so plan accordingly*.
If your saddle isn’t really right for you, you’ll know it. You may find your knees take quite a beating (mine creaked for a few days after one long trip). Having the bike properly adjusted can be critical. Seat height, seat angle, handlebar height, handlebar fore/aft placement, and handlebar angle can all make a difference. A good bike shop will be able to help you with these. Touring style handlebars offer five different hand placements, learn them all. Padded riding gloves (fingerless) can really help.
*Rode from Kansas City to the music festival in Winfield, KS, one September. First day, I realized that although I knew the prevailing winds would be from the southwest, I didn’t think about it. :smack: Fought headwinds and sidewinds for all but 10 miles of a 245 mile trip. Got smart (ha!) and the following year did the reverse instead - rode from Winfield to Kansas City. First day was glorious - tailwinds! First night a norther came in with a system that stuck around for a few days, and I rode the rest of the trip fighting headwinds and sidewinds. COLD headwinds and sidewinds. Had to buy deerskin full-finger gloves that next morning to keep my fingers from freezing off. Lesson learned.