There's a lot of well-intentioned advice in this thread, but most of it is pointless unless we cover some basics first:
1) What size shot are you using? A home defense barrel without a removable choke will always be Cylinder, meaning it has no constriction at all. This is fine for hand-thrown clays (they're called clays or birds, not "skeets" or "traps") as you want the widest pattern you can, but you also need to use smaller shot to have a dense pattern. If you are blasting at the clays with 00 buck then no wonder you're missing. Stick to #7.5 through #9.
2) You need to check your eye dominance. There are two ways to check this, but the easiest I've found is to point at a distant, small object with your trigger hand. I'm right-handed, so I will give instructions as if you were right-handed too. Point with your right hand at the object. Close your left eye. Looking with your right eye, the tip of your finger should be lined up with the object. By contrast, without moving your hand, if you close your right eye and open your left, the finger should "jump" to the right of the object and not be lined up.
If this is true for you, then you are right-handed and right-eye-dominant. Some people are cross-dominant, and that causes issues with shooting shotguns.
As for the gun you buy, I'll throw my cents in with the suggestion of buying another barrel for the 870 you have already. I'd recommend a 28" barrel with a vent rib. 28" or 30" is considered an all-around length for any of the clay games, but it's not because a longer barrel points better. If your stock fits you, you really can't tell how long the barrel is. Not to mention that you shouldn't be looking at the barrel after you call pull anyhow!
No, the longer barrel is good because it puts more weight out in front of the hands which generally makes the gun balance more neutrally and helps smooth out your swing. A short 18" gun is usually too snappy for most shooters.
Cameron
Stand fairly close to the thrower to start with so lead is not as big a factor. As soon as you start to connect your brain will start associating sight pictures with broken targets. Good luck.
