It looks as though everything went fine during the battle. I killed Ulfric with the sword General Tullius gave me, heard his speech, exited Windhelm, saved and exited the game. The Civil War quest line is over. But when I reloaded the game and headed back into the city, first I got a message that I must help the soldiers defending the city before I can enter, but was not actually prevented from entering. Once inside, I found fires and barricades galore. It was difficult to find a way to return to the palace. In the palace, Ulfric's body is on the floor in front of the throne and I couldn't find the new jarl. It looks as though I must proceed to either Whiterun or Riften to unload my loot and spend the night.
Is something wrong? Of course I am eager to buy the house in Windhelm.
I had gone out only as far as the stables. To be on the safe side, I reloaded from that point and continued to Riften instead of going back into Windhelm. A few hours later, I returned to Windhelm, and indeed it is now back to normal, except for the bodies of Ulfric and one of his cronies still lying in front of the throne. Thanks for the advice!
I'm glad I had looted some of the bodies of Stormcloak soldiers as soon as they were killed, because by the time Tullius gave his speech they had all disappeared.