By stoning, I don't mean that if your handsaw is misbehaving, you should go all biblical on it. Instead, I mean there is a way to gently persuade it back to the straight and narrow by using your sharpening stones.
If your saw is consistently listing to one side when you are trying to saw a straight line, the problem might not be with you, but with the set of the saw's teeth.
If it's listing to the same side all the time, that means it is cutting more agressively on that side, which in turn means the teeth have a greater set on that side. You can decrease the set by laying your handsaw on a flat surface and lightly dragging your fine grit sharpening stone along the aggressive side of the saw's teeth. Take light passes and test the saw often. It doesn't take much to correct the cut. By doing this, the saw kerf will be made a bit thinner, so if you don't want that to happen, you might want to enlist the help of an expert sharpener.
Or, play it safe, and just purchase Lie-Nielsen saws.
Last photo added afterwards because one sharp reader noticed that I had used a rip saw to make the crosscuts. (I used my dovetail saw because it is my favorite, it was within reach, and I like the physically smaller size of the saw, compared to my crosscut saw). Nonetheless, the proof is in the photo—both the rip and crosscut Lie-Nielsen saws work great!