A rough and heavy trigger can definitely contribute to shots being pulled. Also, improper grip and/or trigger mechanics will almost always cause shots to be left or low/left for right handed shooters. Many people squeeze a trigger in segments, meaning they take up some slack, rest, take up a little more, stop then finish it off. This last hurrah will cause the shot to go left.
Start by assuring you have a proper grip. With the weaver stace you are using a 2-handed grip (which is good). Make sure the grip is properly placed in the primary hand and the off hand (left for right handed shooters) is cupping the fingers of the primary hand with the thumb pads resting against each other and both thumbs pointing at the target. The off hand actually pulls rearward against the primary hand causing a clamping action which allows the trigger finger (and remaining fongers of the primary hand) to relax. This permits the trigger finger to work independently of the rest of the hand and promotes a smooth and consistent trigger pull that takes up the slack and breaks in one smooth, fluid motion. Here is a link to a webpage that shows the various "combat" grips that are taught and utilized by the top shooters: http://www.handgunsmag.com/tactics_trai
A good way to practice is to get a cheap laser toy like one would use to play with a cat. Rubber band it to the slide of your UNLOADED gun (with the mag out and all ammo in another room). Take your proper grip and stance aiming the laser at a point on the wall and squeeze the trigger. With improper trigger mechanics the beam will bounce as the sear breaks...with proper work the beam will remain vurtually motionless. You can dry fire using this until you see and feel a smooth, fluid trigger pull with the beam staying put. Remember- practive does not make perfect, practice makes permanent...one needs to practice good technique in order to engrain proper methods.
"Shoot low boys...they're riding shetland ponies in search of true grit" Lewis Grizzard