Use a cutting square or other non-slip straight edge to guide the cutter. You can push as in normal stained glass cutting, or you can draw the cutter toward you as glaziers do. In either case, the pressure needs to be even and the speed consistent.
In moving large scored sheets, avoid pulling by the end. The score may run suddenly and not always along the line. Instead, move the sheet with support on both sides of the score. After the glass is scored, you have choices about how to run the score:
One easy way is to move the sheet so the scored line is just inside the edge of the bench. The biggest piece will be on the bench and the smaller piece in your hands. Give a quick, sharp downward push with both hands on the overhanging glass. Having the glass score inside the bench edge gives you a place for the broken off piece to rest, rather than pivoting toward the floor.
You can slide the straight edge under the glass on one side of the score, and press firmly, but not sharply on each side of the score. The glass will break evenly along the score line. This is a more gentle method of breaking the glass. A variation on this is to place a couple of match sticks or glass painting brushes at each end of the score and apply the pressure.
If the glass sheet is of a size that you can hold it in both hands with the score between, you can draw it off the bench, let it hang vertically, and bring your knee up briskly to hit the score line, and it will break easily. This is a showman's way of breaking glass sheets when the score line is approximately centered on the sheet.
Cut running pliers often do not work very well on long straight scores on large sheets of glass. However, if you try this, tapping along the score line before squeezing the running pliers will help the score to run the way you intend. (And defeat the purpose of getting a clean break, as each time you tap the glass you get a ledge on the side of the score line that you tapped). This is sometimes the only way to achieve the break of the score.