You have gotten lots of good advice here, and the good news is that it is all consistent!
Perhaps first and foremost, you need to relax. The only way to really relax is to be comfortable, and the only way to become comfortable is to dive more.
When I first started (and that was only a couple years ago) I had the same exact problems that you describe. When I was trying to do a safety stop, I had trouble staying down sometimes. Part of that was the fact that I instinctively was kicking my fins. Part might have been the weight also, but part was nervous breathing. I was keeping too much air in, and it made me more bouyant. When I finally got more comfortable, and relaxed, I wasn't as bouyant.
When I started I had a real hard time descending. Part of it was the air trapped in my wetsuit, and part was the nervous breathing again. The biggest, stupidest part was this: as an instinctive move, every time I wanted to descend, I would take a huge deep breath of air. That is what you do when you are just simply swimming (or maybe snorkeling) and you want to go under water. For scuba it is exactly wrong. Once I realized, I was able to control the urge. I don't need a big lungfull of air when I descend, as I have a source on my back. Now I simply exhale deeply, and it is a lot easier to descend. No more working really hard to get under.
A friend of mine has trouble descending because she always wears a hood. Once she found that the hood collects air, she has found that if she smushes the air out of her hood when she descends, it makes it a lot easier.
Anyway, these are just some other things to consider. The other advice was all good, but these were just some things that I have seen or experienced that no one else mentioned. I hope it helps.
Relax, dive within your limits, get more comfortable and relax, then expand your limits as you get more comfortable. The cycle si great!
my dos centavos, and only because you asked,
Wristshot
PS: OKron, my buddy had trouble with a weight belt so he bought the DUI weight harness. He couldn't get a weight belt to ride right; it was always falling down (no hips). He absolutely loves it. He has some weight in the integrated pouches of his Ranger, and a bunch in the DUI. It works rgeat for him.