Interesting concept. The problem seems to be, though, that the US military has been largely a force raised for emergencies and then more or less completely disbanded. Actually, there is one possibility - during the Civil War era, most of the units used were based on the state level - the 1st Michigan, the 2nd Maine, and so on. However, some units were proper US Army units - they were the minority by an order of magnitude, but the point remains that they existed. On the last day of the Civil War, when victory was already won, the various state units were transformed into proper units of a real national army, their state designations removed.
The POD seems to be, what if that doesn't happen? If so, it could be that the real US Army forms an elite core of the military, as a well-trained standing army, while the various state units - perhaps something like the National Guard or something - form the bulk of the US forces, but retain their identity as being associated with one state or another, and as part and parcel with that, are not part of the standing US Army. Which would, of course, mean that they would not be as well trained as the real US Army.
Hm, now I've got a vision of the 20th Maine fighting the Germans in the trenches of a Great War analogue. Could work, perhaps?
What that could also lead to is that in something like the present day, the "regular army" could fulfill a sort-of Special Forces role, while the state troops are the normal sort of troops. Or at least, the "regular army" would be better trained and equipped - the exact thing you wanted.
Of course, something like this would have another interesting dimension - the results of battle losses would be less evenly distributed, so that depending on where in the line the fighting was fiercest, men from different states would die. Could be problematic - imagine what the people of a state would have to say if, for example, one of their units got chosen for some suicidal assault. Or if the 1st Rhode Island got cut to pieces in a battle of that WWI analogue I mentioned...
Hm, well, those are my thoughts at least. What do you think about the plausibility of that scenario?