If taking a dart out works well with sleeveless garments, should I also be making this adjustment when I sew garments with sleeves? If so, would I have to alter the sleeve a bit as well?
That depends on the sleeve, but probably yes.
If the sleeve is set smoothy into the arm scye you will need to make a similar 'dart' at the same level in the sleeve cap. If the top of the sleeve has gathers there's no need to adjust the sleeve, just gather as much as needed.
The other option for anything with a sleeve is to use a shoulder pad (or a thicker shoulder pad) on the low side to even out your body shape. In that case you probably don't need the 'darts'.