Sherman was not facing a Confederate army on his march and didn't fight a pitched battle.
Lee's mentality after a victory was not to go home and wait for the next battle, but to press on.
Then why didn't he press on after Chancellorsville?
He would have sent them to Virginia.
He had no supply line established.
Which he would have replenished from Union stores.
Union stores where?
Lee was facing an army, and was able to gather quite a bit of stores locally despite that fact.
Then why didn't he press on after Chancellorsville?
He did. He went to Gettysburg. Within a couple of weeks of Chancellorsville he was already advancing northward. Brandy Station took place on June 9th.
He had no supply line established.
Indeed he did. So well established, in fact, that his troops had time to capture and send to Virginia black Pennsylvanians as "contraband." If he had the resources to kidnap people and ship them South, surely he could have used those same resources to send wounded men along the same route.
Union stores where?
The Allegheny Arsenal (still being restocked), Harpers Ferry (already worked once) and the Washington Arsenal.