What arrows are you shooting ? Use a leather glove for the bow hand on the bow with no shelf. What 's wrong with either bow to hunt with ?
With the York bow , the arrows has to be spined so it can bend around the bow as you release. With the Quillian if it's more center cut, the arrow doesn't have to bend around the bow as much, so you might be able to use a stiffer spined arrow, unless they are already flying well.
P.S. Don't be over bowed if you are just starting out, a lighter draw weight will allow you to gain good form and then move on to heavier draw weight bows.