When the Germans used Cannae, they didn't refer to a fixed scheme of operations; it was shorthand for the goal of rendering the enemy's main field army combat ineffective. They thought the best way to do this was through encirclement and concentric attacks, but aiming for a blow by blow replication of a classical battle was unrealistic to say the least, and they realized this.

Regarding the American Civil War, the most ardent devotee of the battle of annihilation was of course Robert E. Lee. Trading man for man in a grinding attritional struggle was a losing game for the Confederacy, and submitting to a siege was a recipe for almost certain defeat. As a rule, Lee sought to destroy the main Union army through concentric attacks.

Should probably define terms before continuing. Hannibal's army, his Roman opponents, and most Western armies up to the end of the Seven Years War were essentially unitary linear armies. What this means is that since it was only the complete field army that had all the combat arms to fight a battle, all its troops had to march together along the same route to the battlefield. It would then deploy into its battle array essentially facing in one direction; cavalry could outflank the enemy battle line, but infantry could do so only with great difficulty.

Towards the end of the Seven Years War, and most resoundingly in the Napoleonic Wars, armies experimented and implemented smaller combined arms formations, like corps and divisions. Because they had all the arms necessary to fight a battle, they could march separately, but remain close enough to support each other on the day of battle. This separation, and the act of joining, made it possible to translate the force's approach to battle into tactical maneuvers on the battlefield. Imagine an army of three corps; with infantry, cavalry, and artillery, each is essentially a small army, and even a superior force is going to take a lot of time to destroy it. The corps march, say, north, side by side; if the center corps bumps into the enemy army, then it's supporting corps are approaching from the east and the west. On the battlefield, these approach marches become massive concentric flanking attacks, one of the best tools for routing an enemy army. Thus, engineering a large pincer was much easier in the post-Napoleonic

This is essentially what Lee attempted in most of his battles. In the Seven Days Battles, he had Longstreet and Hill attacking from the West while Jackson's command came down from the North, with the aim of cutting off and destroying first Porter's Corps, then the whole Army of the Potomac. At Second Bull Run, while Jackson's command held off Pope's Army of Virginia attacking northwards, Longstreet's command attacked up from the south, scattering the Union army. During the Maryland campaign, the Army of Northern Virginia split into several columns, and no fewer than three converged on Harpers Ferry to destroy the large Union garrison there.

Now, I'm not going to list every single one of Lee's battles, but you get the idea; seeking destruction of the enemy's main field army through concentric attacks, typically by separated portions of the army.

Regarding Sherman's campaigns, the intention with the 1864 campaign was not the massive envelopment of Lee and Johnston together it eventually became; U.S. Grant's memoirs describe it as being primarily aimed at destroying Johnston's army and capturing Atlanta. What happened though, was that the main Confederate army in Georgia under John Bell Hood disengaged to try a large scale turning maneuver, leaving Atlanta and the Georgia interior unprotected. Disregarding the threat to his supply lines and living off the land, Sherman marched on Savannah and made it his new base of operations. From there, he could proceed north along the Eastern Seaboard against only light opposition (until the Confederates were able to shift the Army of Tennessee from Mississippi to North Carolina after Hood's failed Nashville campaign). This threat of envelopment forced Lee to abandon Richmond; the plan was for the Army of Northern Virginia to link up with the Army of Tennessee and win a battle of central position against Sherman's army before turning to Grant's. However, during the retreat from Richmond, Lee's army was encircled in Appomattox Court House and surrendered.

Most battles had in one way or another an attempt to recreate a 'cannae' by flanking or encirclement, otherwise fighting would be reduced to two armies walking into eachother

The rebel army had a chance at a recreation of Cannae at Gettysburg but poor coordination and communication prevented that (As a matter of fact, two separate contingents of the rebel army had marched into the rear of the federals twice, but countermarched back to their designated positions to launch attacks at Little round top and Culp's hill respectively)