I understand the anxiety -- I'm sure we all do. I still experience it myself even afters *years* of using a tarp, because I only manage to get out into real weather about 1-2 times a year (mostly Texas nights are pretty mild, except for the odd big storm here and there).
That being said, I'll suggest a couple of things, which have already halfway been mentioned by others:
1) Using a tarp will hone your site selection skills very quickly, and by necessity.
I just completed a trip in Indian Peaks, outside Boulder, where I was the only one with a tarp (a Trailstar) in our party full of 2-man double-wall tents. I chose sites that showed signs of good drainage, e.g. didn't have patterns of dirt scouring or bare patches of ground in which water could pool. I chose areas of faint local maxima, in terms of elevation, aka sites from which water would run away from, rather than pool into. If that isn't available (and it often isn't), then areas with short hardy-looking grass are my favoritem because excess water will tend to soak in there, or at least run down near the roots of the grass. In these areas, you and your gear are in some sense suspended ever so slightly above ground level by the many blades of grass or leafy ferns -- and that small difference makes for a campsite that stays much drier, cleaner, and more comfortable.
On that trip, the other tent users with their waterproof floors and footprints just tended to choose the flattest, cleanest areas on the ground, which usually were highly scoured areas with no grass, aka where other tent users had planted down 1000 times before. When we got a lot of rain on the last night, we were lucky to at least not have any leaks into the tents, but they DID get a lot of mud splashing up inside their outer walls, onto the mesh, and onto any gear that was stored in the vestibules.
My site, on the other hand, was pristine. When I struck the Trailstar down in the morning, about 96% of the area underneath was totally dry -- the only space in our campsite that was so, after the previous night of storms.
Point is, as others have said -- if you learn just a few tricks for site selection, you can end up having a MUCH better night under a tarp than under a tent. That being said, the Trailstar was not my first tarp, and my earlier experiences were a bit less pleasant. That brings us to point #2 ...
2) I agree with others that you may want to start with a larger tarp. More than that, a SHAPED tarp will make you feel warm and safe and protected, while still being way lighter and way closer-to-the-wilderness than a double-wall tent, aka a portable cottage. Just kidding, I haven't reached that level of snobbery yet. But I'm close.
There are a few good entry level options there that are not absurdly expensive. Off the top of my head, the first is a GoLite SL2, which is nice and cheap direct from GoLite, and should be plenty protective and comfortable for one person (from what I understand). Other options on which you may be able to score a discount or a used Gear Swap deal -- a pyramid from Black Diamond, MLD, or Oware. They pop up fairly frequently on the web or in these parts.
The fun with a shaped tarp is that you've got real protection in case you're trying to really prepare for several days of serious rain, and can't get that out of your head. In the parts where I hike most, I almost never encounter that kind of climate, so I can't empathize or advise you 100% there. All I'm saying is, as noted in my story above, a good roomy 2+ silnylon tarp, weighing about 1-1.5#, can be *way* more comfortable and enjoyable than a 3.5# double-wall UL tent that has likely less floor space. I dunno if 3.5# is the current industry standard for those, because I don't buy them, but you get the idea.
3) As a bonus or piece of insurance, you could easily consider a cheap and light bivy to go around you and be your waterproof floor. I use a Titanium Goat bivy with a water-resistant / breathable top and a silnylon bottom, and it weighs about 6oz. When I've regretted pitching my Trailstar a little too high and breezy, and the weather comes in harder than I expected, I'll just zip up my bivy against the spray and light splashes that might (rarely) come inside the tarp, and otherwise sleep the night away happily.
So, these are my recommendations. In summary, I think that if you can combine 1) good site selection, 2) a roomy shaped or pyramidal 2-person+ tarp, and 3) optionally a lightweight bivy; that ultimately you'll find the transition not incredibly taxing. And you almost certainly won't look back.
That is .. unless your climatic conditions are way different from what I've experienced in my (very limited) time.