I should be moving my my grip back the other way a little, pushing my thumb forward and my (K)nuckles back to the front of the bow.
Yes, that sounds right; but without the benefit of pix, it's hard to say.
Ideally, you want the left edge of the grip aligned with your "life line" (RH shooter) which would place the bulk of the grip in the meaty portion of your hand at the base of your thumb. In other words, the 'belly' side of the grip would be sitting between the line at the joint/base of your thumb and the next line, your "life line". This will allow your relaxed fingers to curl loosely around the back of the grip.
If you have trouble with string to forearm/wrist contact, you might try opening your stance a bit.
If you've been using your current grip for some time, it may take some dedicated time on a blank bale with your eyes closed focusing on the new grip, until it happens automatically and feels normal to you.
Good Luck, and Keep it Fun!