I'm currently reading "The Story of the Confederacy" and while I didn't come across anything about armed conflicts between states, there were definitely a lot of political issues caused the Confederacy's decentralized structure.

Just as the southern states resented the Union because of its strong central government, so too did a lot of states come to resent the Confederate government when it tried to exert control over the states to unify the war effort.

A great example came towards the end of the war, when the governor of Georgia gave a furlough to all Georgian troops to go home and tend to their crops. Naturally, the military situation for the Confederacy was falling apart at that time, and it was probably the worst possible time for a single state to unilaterally decide to stop contributing to the war effort.

In addition, there was a lot of friction caused by the practice of some states deciding to give generous deferments to their own men to not serve in the military, while other states were more strict about it. This created the perception of lopsided commitment to the war effort. A lot of states also tried to argue (and some times successfully) that their own troops should be kept within their states borders to defend their homes, instead of serving in a centralized Confederate Army.

Overall, the Confederacy's decentralized structure made it exceedingly difficult to coordinate the war effort for their national government. They couldn't effectively mobilize their manpower since the states had too much control over it, and often acted in their own best interests instead of the nation's as a whole.