First thought is it could be a "plucky", sloppy (no offense intended) release, resulting is a lot of sideways vibes as the arrow releases.
Second, have you tried adjusting your string twists to fiddle with brace height at all? Could be an improperly set brace height. Check the recommended BH for your riser, then add string twists to get the BH into about the mid-range.
Third, that sounds like what my wife's first bow did, when she was shooting 18# limbs at 24" draw length. 24# limbs are perfect for learning, but on a 70" bow, they're (for lack of a better term) wimpy, so that there's not as much tension at rest to stop the "wobbling" you see after the shot. All bows would theoretically do that some, but the heavier the limbs, the sooner it'll cancel. At lower poundages, the vibrations can last longer.
As for the arrow flight, how well are the arrows matched to the bow? In other words, what level of preliminary tuning/set-up have you done? If the nock height is off, the plunger centershot is way out, your follow-through isn't too clean, or the arrows are just WAY too stiff (or weak) for your DW, it could be causing some really poor arrow flight.
These sorts of things are really tough to diagnose textually, but with a bit more info, we may be able to help you get closer. Also, more info on your background (is this your first bow...?) would help. It doesn't sound like you've got anything serious to worry about, and it's something that will work out as you get your bow set-up and you develop your form more.
