Don't sweat the bullet issue too much. Though your NH whitetail are likely larger than what we see in much of Texas, my experience has taught me you don't need anything special to cleanly take deer at your distance of 50 yards or less. I've successfully used 125 XTPs on a number of them, but bigger deer would probably make me want more weight. If you stick to a decent 158 gr. slug in the vitals, the deer should expire pretty quickly. Shoulder shots would need a stronger bullet, but still wouldn't need anything magic. I think Ernie's advice is right on track. The main ingredient in handgun hunting is accuracy. It's a lot of fun and hard to go back to a rifle! Good luck!
But for the grace of God, there, I go.