One way to deal with the shortage is using Armies as fighting stacks. Another is to recognize that Virginia is crowded with 3 stars. Once Lee enters someone can go West. I usually send JoJo to take over for ASJ in TN/Kentucky, who I then either send on to Missouri or to NO. Bragg is a pretty good Corps commander, don't sideline him (I forget if he auto-promotes or not in the GCs, he is a 2 star in most of the shorter scenarios.) I also minimize the number of Armies I create, since I sometimes need three stars to serve as corps commanders.
Once things get going for me I do get a lot of promotions, especially if Athena tries to break through in Virginia, but there is definitely a point in every game where I have not had enough decisive wins and feel the leadership crunch. I usually take it as a sign to slow down on the building a bit. Also, focusing more on artillery builds reduces the number of commanders you need overall (it packs more punch in less space and you won't be able to afford to overbuy on infantry if you are buying artillery).
If you are trying to push one of your early game all-stars to Army command, do it with Longstreet, he improves marginally as he goes to 3 star (although he loses a good ability).