Win73, either of those loads should be fine from your little NEF.
A) Putting down injured animals and B) shooting frisky, healthy ones in their own element is two different things; I have used handguns to do plenty of both.
I can honestly say that the one deer I shot under B was an utter failure rooted in my own inexperience at the time. I shot way too far (hit both times) and used the much-touted 110 grain ammo of the mid 70's. I've since seen several deer shot with hot SWC loads and I wasn't impressed; but again none of those were perfect shots.
If I was deer hunting with a 357 today, I would load it with 180 XTP's, running ass fast as prudence allowed, and I wouldn't take any raking shots. I'd aim for the heart or the spine at the shoulder and I wouldn't shoot past 50 yards, period.