I hope your finger recuperates soon. You didn't tell me which one before. I assumed it's the index. If so, you can always make do with your middle finger on the trigger & bumper.
It's the middle finger on my right hand. At first doing ust about anything with that hand hurt, so I spent a lot of time playing WoW one-handed with just the keyboard, and using the mouse in my left hand for NPC interaction. It's been 10 days now and the pain is mostly gone, unless I do too much with that hand. Thumb+index finger on the controller is fine, but the mouse is still a problem. I can't use my middle finger for right click, and using my ring finger to do so eventually causes my hand to cramp up. (and if you're using the mouse to turn on WoW, you're holding the right mouse button down almost constantly.)
Mmm, that doesn't sound good at all. Everything else doesn't matter if the gameplay is shit because of shitty input handling. Keep me posted on this, because it sounds like a deal breaker.
Edit: Try using your controller in wired mode (USB to mini-USB cable), and see if there's any improvement. According to some things I've read, it can make a noticeable difference (apparently more than the 9ms lag claimed for wireless mode--might be an OS thing?).
First of all, I remember the Xbox 360 version feeling a bit sloppy too, and played the single player stuff with Assisted Aiming mode. So it's not like they messed it up when porting it. I was just messing around with the game and here's what I noticed. This is all from feel, so I could be wrong on some of these points, but generally I'm pretty nitpicky about responsive controls. Wired vs wireless made an almost imperceptible difference. The deadzone seems ok, maybe a little big, but not as bad as I initially thought. I think it's the smoothing that's giving it most of the sloppy feel. You don't really notice it in third person as much, because when running around or driving you don't mess with the camera as much. When I aim my weapon or go into first person I start to notice it though.
Turning off first person head bob helps a bit. Turning up the sensitivity felt a bit better too. Strangely enough, I think changing my Targeting Mode helped too. I swear when I put it on Free Aim it feels like it's snappier and easier to control. The other modes snap to targets when you aim, but then are sluggish when you try to adjust. I feel like adjusting the FOv down a bit might have helped a little as well, like maybe it's related to performance? I know when I play on my PC with v-sync on, if my PC can't maintain enough framerate, my mouse begins to feel sluggish and unresponsive.
By the way, from an options standpoint, they went above and beyond what most console games give:
Allow independent camera modes: it remembers which camera mode you last used in a car or on foot. I use this to go FPS mode on foot, but drive in third person.
HUD: on/off
complex or simple (dot) reticule
size of the simple reticule
depth of field: on/off
third person look sensitivity
third person aim sensitivity
first person look sensitivity
first person aim sensitivity
first person FOV slider
first person ragdoll: on/off (do you want it to jump to third person when your body ragdolls?)
first person combat roll: on/off (should the camera roll when you do?)
first person head bobbing
first person third person cover: on/off (jump to third person when you take cover?)
allow movement with sniper
driveby control type: aim+fire or just fire
Also remember that most modern TVs have all kinds of post-processing. See if there's a "gamer mode". My Samsung TV from 2010 has one. It turns off all the post-processing and reduces the delay.
I'm going straight from the Xbox One to an ASUS 2ms LCD monitor via HDMI. No receiver.