Coldmold
I have done quite a bit of deer hunting with handguns, and yes, in my opinion, a ,357 mag. with a good quality expanding bullet like the XTP will work, especially on lung shots. Heck, I once killed a black bear with the Remington 357 125 grain hollowpoint police load, one shot and down.
However, I would seriously consider staying with your 44 mag using a hard cast semi-waddcutter type bullet, 240 to 270 grains in weight at about 900 to 1000 fps. Recoil is not bad, and this load will give plenty of penetration, cutting a nice hole thru your deer, with a good blood trail. I used to load the same bullets at 1200 / 1400 fps, but have noticed no real difference in killing power compared to the slower loads, just less recoil and blast. Most of my shots are 50 yards and under. Another option might be a 180 grain jacketed at about 1200 fps. Should give good expansion in the lungs, just stay away from heavy bone / raking shots.
Larry