Getting weighted correctly and maintaining neutral buoyancy aren't really the same thing.
Being correctly weighted makes it easier to adjust the BCD for neutral buoyancy, but you will still have to adjust for the weight of air that you have consumed and for the change in buoyancy of your wetsuit as you descend and ascend. Just like having the right weight makes life easier, so does having a thin wetsuit, but obviously if you are diving in cold water, then you are stuck with a thick wetsuit and the large change in buoyancy with depth changes.
Your buoyancy will also change around 5 pounds as you breathe in and out, so what you really need to find is the right amount of air in the BCD such that you are neutral, on average, over each breathing cycle.
Most new divers spend most of their dive significantly negatively buoyant. They fin upwards at an angle to maintain depth and will sink if they stop finning. At the beginning, the most essential skill you need to learn is how to stop finning for a moment.

If you sink rapidly, then you are very negatively buoyant. Start finning again and add some more air. Eventually you will get to the point where you will rise when you breathe in heavily, and descend if you exhale fully.
The effect of breathing has kind of a delayed reaction on your depth, and it takes a while to get used to how inhaling a lot will add a bit of buoyancy and stop any downward motion, and then begin to send you upward. In the same way, if you wait until you are moving upward and exhale, it will take a few moments before you start to descend.
The safety stop at the end of the dive is a good place to practice maintaining depth with no finning or arm waving. If you are horizontal, you will have more resistance to up and down movements and staying at a near constant depth will be easier.
Using a frogkick rather than a flutter kick is a tremendous aid to staying neutral. A continuous flutter kick can hide changes in buoyancy. The long glide phase of a frogkick highlights any buoyancy change.
You can make relatively large changes in buoyancy simply by spending more time with your lungs near full vs spending more time with your lungs near empty. I find that my primary, short term, buoyancy control is by my breathing pattern/fullness of lungs. That keeps me neutral, and then I adjust the BCD so that my breathing pattern is more or less centered.
Again, the key is to simply stop all movement and adjust the BCD so you average out to the same depth over each breathing cycle.
Charlie Allen