If you find yourself having trouble controlling buoyancy during a shallow dive, it is likely that adding weight will make things worse, not better. In this case, I am talking about people finding themselves floating to the surface during the dive, dumping air to go down again, adding air, floating up again, etc. You see that a lot in shallow reef areas like Molasses Reef in Key Largo. I am not talking about having trouble staying down with a nearly empty tank after doing fine before that while the tank was more full.
When you are diving, you achieve neutral buoyancy through a balance of whatever is pulling you down and whatever is pulling you up. You are able to vary your buoyancy during a dive by changing the amount of air in the BCD and by inhaling and exhaling. If you have roughly the right amount of air in the BCD for your depth, you can control your buoyancy easily simply by the way you breathe. If you ascend enough and the air in your BCD expands to the point that you can no longer control your depth with you lungs, you have to dump some air. If you descend enough for the air in the BCD to contract too much for your lungs to control it, you must add air to it.
The deeper you go, the easier this is. That is because the amount of change in volume of the BCD by ascending or descending a few feet becomes less and less. Because that change is so pronounced in shallow water, it becomes most difficult to overcome those changes at those depths.
The less air in the BCD, the easier it is to control. A properly weighted diver will have very little air in the BCD. For every unnecessary pound of lead you wear, you have to add about a pint of air to the BCD to achieve neutral buoyancy. Every unnecessary pound thus adds more air that will be expanding and contracting with every little change in depth. Lots of people dive a few pounds heavy, meaning they might have an additional quart or so of air to contend with. That's not too bad. Add much more than that and you will soon find yourself at the mercy of a constantly expanding and contracting air balloon in your BCD.
When I am instructing in the deep end of a pool (about 12 feet), I am a few pounds overweighted intentionally, as most instructors are. I do a demonstration in which I go from the very bottom of the pool to the top and then back down again using nothing but the air in my lungs to control the depth. No problem. On the other hand, when I am practicing for technical diving in the pool and have big double steel tanks (etc.), I am significantly overweighted. I can only control my depth with my lungs over a range of a few feet because I have to have so much air in the wing that it has become beyond my control.