Being vegan, I was BEYOND grossed out by the whole thing at first. I think you should do whatever helps you feel more comfy about it.
I have one side of my 2-well sink that is ONLY for raw meat. The other side is everything else. I have a cutting board, tongs, knife, and poultry shears ONLY for the meat. When I'm done w/ prep I spray the sink, my tools, and my hands (after washing) with a vinegar/water solution (which is what I use for general cleaning). When I started I did wear gloves, but honestly they're more trouble than they're worth.
It DOES get easier. I hadn't eaten meat in 25 years when I started taking my dogs and cats raw--I literally had a meltdown in Whole Foods and had to call a friend and have her talk me down, LOL. Just talking w/ butchers in the meat department made me want to cry.
But now I'm pretty blase about it, so it WILL get better. Just hang in there!!