Nothing wrong with the .243 for a deer gun. With the right 90gr. bullet, it can maintain 1,000ft. lbs. of energy at 400 yards... still at the set suggested minimum for whitetails. I don't really buy that, as I've killed with far less energy, but it's a mark to go by.
The .260 is my favorite deer rifle. It has the power and trajectory along with good ballistic coefficient with hardly any recoil.
I'm a Remington 700 fan. Their accuracy are noted well and many target guns are built from their actions. However, I've been noticing a lot of people are happy with their Savage guns being accurate right out of the box. I think you'd be happy with a Savage .243. It should give you all the accuracy you'll ever need chasin' whitetails!
Also, if you can devise some kind of patch or blocker for your left eye when you shoot, it is much better than closing or squinting. You lose visual acuity when you do that and it gets worse over time.
Good luck,
iSnipe