It was all a dream. You character needs to wake up.
No, not really. One of the best things about TES lore is the concept of subjective reality. The act of donning Sheogorath's regalia was as much symbolic as it was physical. By bearing his staff you became Sheogorath. His plan was for you to rebuff Jyggalag to preserve His realm, but as it turned out Jyggalag had planned for the exact same thing. The difference was that Sheogorath had planned to strip you of His power and reclaim it once the Greymarch was defeated, but Jyggalag was pleased to let you keep it so He could take all his soulless toys and go home (which is to say, nowhere, for He is His home, since Prince and Plane are one and the same).
It doesn't matter that there is a new Sheogorath, since He will be exactly the same as the "old" Sheogorath. A feeble mortal soul cannot withstand the power of an entire Plane of Oblivion, and over time the Champion would assuredly have more and more trouble returning to the mortal plane until it was impossible except under specific circumstances. By that point his personality will have been crushed completely, and some other mortal will have to use a dinner fork to kill a giant netch in his stead.
This debate will be proven one way or the other in Skyrim if we see Jyggalag alive and well or not. If Jyggy is out and about trying to Order everything up to make it all crystally and grey, then the Champion has been transformed. If he was just an illusion then Sheggy had his fun with a mortal and all's back to normal now.