Try this with a clear weapon. During your dry firing exercises, after you pull the trigger slowly let off the trigger. You will feel (and sometimes hear a click). That's the firing pin resetting itself. That is as far as you need to go. You can pull the trigger for the second shot. If you practice this you will be less apt to snatch the trigger. Slow, steady constant pressure on the trigger.
As for the grip, that is something I have found to have fixed with practice. You don't won't to choke it but you don't want a light grip( I have seen pistols not cycle because of a loose grip). Remember your hand is part of the mechanics. The gun wants to go backward when fired. Your hand is the backstop that stops it and allows the slide to cycle properly.
Glocks are easy to learn on (I have 3 myself 35,33,22)
The internal safety gives them a long trigger pull. Sometime the anticipation as you are slowly pulling the trigger make you want to snatch the trigger. Once you get past that and become comfortable with the shot, the rest is cake.
By, the way Raleigh. I shoot IDPA in Granville County as my work and honey-doo schedule will allow. There are a few IDPA pistol matches in Raleigh.