I'll tell you exactly what's going on.
spenoza, in the first Dungeon Explorer as you may or may not have noticed all your characters have different stats, one of them being "agility." In the first game, your agility rating doesn't seem to have any noticable effect on how your character actually controls.
They fixed this in DE II, so your agility rating directly affects how your character controls. When you start out, obviously your agility is at it's lowest and it causes the control to feel "drunk" as you put it. As you progress through the game and your agility improves, so does your control correspondingly. At some point your character will control as well as any character from the first game, and then even beyond as you continue to become more agile.
As you may or may not know, the color of the "crystal beans" you pick up after defeating a boss (or changing characters) increase the stat of the corresponding color. If you wait a moment instead of picking up the crystals right away, they will eventually cycle through all the colors. I don't remember what color corresponds to agility offhand (light blue, maybe?) but you can increase your agility quickly this way.