Some good ideas here already. I like 10 Wheels tent/pannier suggestions for a first tour. That's the 'start modestly and work your way up' approach. Nothing wrong with that. Others like to buy the very best up front. Just depends on your philosophy and bank account. If you're pretty sure touring is going to be your 'thing', might be better to go for quality up front. You can always sell it for about half what you paid, maybe more.
I'm a big fan of the
Eureka Spitfire 2. It's a double wall tent, especially suitable for use in hot, humid weather, if you don't mind leaving the fly off for breezes and star gazing. The Coleman is single wall.
I use a flannel sheet my wife sewed up for me for summer camping. On a full length Thermarest Prolite. The pad is not a place to skimp. More comfortable is the NeoAir, but $$s and takes a lot of air to inflate.
If your bike is in good mechanical condition at the start, chances of significant trouble are minimal. Tires with a puncture resistant belt are pretty much a must. I like Maxxis Overdrive 700x38. The Schwable Marathon series is the most popular, and usually the most expensive. Vittoria Rondonneur and Continental Travel Contact have their share of fans.
Spoke breakage can be a problem. I always carry a temporary
fiber fix spoke. Have used once in 6 years.
Carry 4 liters of water, refill at every opportunity. More if wild camping. Adjust for refill availability.
Saddles are very personal. What works for one won't for another. The B17($$)is a huge favorite, but requires a long breakin for most to be comfortable. And rain is not its friend. Been there and done that. Ride a Velo Biologic now, but not very comfortably. So does the Dallas Police Department bicycle patrol.
To stealth camp, pick a secluded non-fenced, non-posted spot, don't be seen entering or leaving, pitch and spend the night. Practice 'leave no trace.' Not hard to do, but a bit more stressful than public camping. I've done a good bit and never been disturbed. I differentiate 'stealth' from 'wild.' Wild being in semi public view, but where no one is likely to disturb you, and done without specific permission. Behind a rural church would qualify. A cemetery would qualify. 99.5% of people have no interest in approaching someone who would do anything that crazy. Common sense will carry the day.
I know your original destination was NO, but should you change your mind, head west, north of I-20. Several conveniently placed state parks in La., and in Texas. Good roads mostly, 'cept for the rough Texas chip seal. Or north thru Mer Rouge and get you some of their famous ice cream
Msg me if you would like more info on routing west of Monroe.