May I suggest snap caps and have someone load your magazine mixing in the snap caps among the live ammo.
Of all the range exercises I've done, and all the reading, and all the dry-fire ... ball and dummy drills (snap-caps randomly in the mag) have produced more dramatic improvement in my shooting than anything else. Once you can pull that trigger every time as if it was on a snap-cap, you're going to get tighter groups, move closer to the POA, and build valuable muscle memory. Flinch all you want AFTER the shot goes off, it won't affect the POI much. You can work on follow-up shots after you solve the flinch on the first.
As to your "limp-wristing" ... get a good weaver or modern isosceles stance with the 2-hand push/pull, and grip the pistol like you're keeping someone from taking it away from you, using 9 fingers. Strong hand grips first, weak hand clamps down on top of strong hand. Do some healthy dry-fire in this position with the nice new snap-caps you're going to buy for the above ball&dummy drills.
(My snap-caps look like crap, by the way. They were shiny and new only 4 months ago, and in that time they've been chambered/extracted uncountable times, dropped on the concrete floor, stepped on, ridden around in the range box, been used for function testing, etc etc)
I've been using a beat-up tennis ball for a grip exerciser, gripping with three fingers until I see a tremble, and then
smoothly working the trigger finger. You can do the same with your steering wheel as you drive. Actually, you can do that exercise with just about any squeezable object in your hand. Strengthening the grip will help you ride the recoil and maintain your hold, rather than the gun working into a new position every shot.
(I bet you needed to adjust your grip frequently while shooting ... that's why, the gun wasn't staying clamped in your hands)
Good dry-fire basics
HERE (sorry for the pink ladies' page, but it is the best dry-fire primer for a new shooter that I know of) Actually, if you haven't, go read cornered cat's entire site ... she doesn't update much, but there's a wealth of info there for new shooters of any plumbing style.
ribbit, you're now more proficient than some law enforcement professionals or active-duty military ... and better armed than a huge fraction of the population.
Don't panic, keep shooting at a level that challenges you, and if you don't have one, get a .22 pistol that mimics your desired defense pistol. If you don't know what your desired defense pistol is yet, get a .22 anyway! (I like Ruger rimfire pistols, but get whatever fits your hand best)