I have known just a few in my forty plus years of shooting a bow that were true snap shooters. They were good, very good. Having said that, it is the exception rather than the norm.
It would appear that you are not getting the good back tension that is a desired effect regardless of the method of shooting a bow.
I find, just my opinion, that release problems are problems that seem to go away when we do the other things we should do in the shot process.
We don't release the string, we loose the string. One is an act of trying to get rid of the string, the other is letting the bow limbs take the stirng from our fingers. It is just as simiple as relaxing the back of hand.
It will happen when we put total focus on the what we are shooting, slightly pushing on the bow, toward the target we want to it, and keeping everything lined up. That is to say, the elbow of the drawing hand, with the arrow, which should be in alingment with the spot we want to hit. If you do those things, release problems will vanish overnight.
Regarding snap shooting, I too am an instinctive shooter, who in recent months have purposely began to hold longer in the shot.
This has increased my ability to hit much, much better. I do the same thing I did before, I just do it in three seconds as opposed to the one or two seconds before.
I practice a lot, hunting gear, hunting conditions. I have killed four of five game animals that I took running shots on. I don't practice running shots alot. You will adapt to the hunting situation.
I am concerned that you may start to develope what we call, "target panic' or "freezing" which is the inabilty to come full draw. It can be best avoided by not shooting bows too heavy, and coming to full draw with good back tension, or positive pull through.
Much has been written in this forum about such things, I am sure you can find them. I hope this helped, welcome to the ranks of traditional bowhunting, and if I may be of service please drop me a line.