Yes the gun moves before the bullet leaves the barrel. When you think about it, the gun starts to move the instant that the bullet starts to move. For every action, there is an equal and opposite reaction. You can't get around the laws of physics. Heavier bullets have more mass, and cause the gun to move more in the opposite direction (within reason of all other things being fairly equal that is).
I don't use bipods for several different reasons, only one of which is the very likely chance that it will throw off the accuracy of the gun. It can change the pressure point on the forend, or can change the "harmonics" of the barrel, as well as change the way the gun comes back during recoil. The sooner you can adopt a "regular" ie consistent method of hold, the sooner you'll be able to shrink your groups. Are you primarily shooting at a range, from a bench, or are you primarily trying to dial in a hunting load for use in the field?
I have used blocks of wood, sandbags, front rest only, and front rest and rear bags while at the range. While in the field, I have used a walking stick, a backpack, a hat, a tree branch, or my knee as a rest. Sometimes you have to improvise in the field. Whatever kind of "rest" I'm using, I try to make contact with the gun as far back towards the frame as I can.