I think people somewhat overstate how important the Baarova affair was in the greater scheme of things. What really pissed off Hitler was that Goebbels was, in his opinion, not warmongering enough. Yes, you read that right. At the height of the Sudeten Crisis Hitler had a big military parade in Berlin. But the Berliners were sullen and unenthusiastic because they feared war. This annoyed Hitler, and he blamed Goebbels for not having whipped them into enough of a frenzy and said that his propaganda had been too peaceful. Goebbels proceeded to try and fix this, and later whipped up the SA to burn down synagogues and murder Jews, which also served his goal of trying to further radicalise the at the time rather stagnant anti-Jewish policy, but the damage was done. Plus 'Kristallnacht' ended up with Göring and the SS taking a leading role in anti-Jewish policy and taking things into a less 'loud', more insidious and bureaucratic direction, so he also didn't get his way there.
Goebbels' standing didn't recover much in the early stages of the war because there's not much need for propaganda when you mainly have victories to report. The main point of the much-misunderstood 'total war speech' (which was not about 'let's have a forever war', contrary to what certain Hoi4 mods say, Goebbels, while a Nazi zealot, was not a super militarist) was to get into Hitler's good graces and position himself as someone who could really mobilise the German population and run things on the domestic front. It's worth pointing out that said speech failed at everything. As Longerich shows in his monograph about it, the reaction of many German citizens was cynicism, and Goebbels was so disappointed that he didn't do much on the total war propaganda front until he got his Plenipotentiary for Total War gig following the failure of the 20 July Coup. As for Goebbels having an accident, that's unlikely. The Nazis rarely killed their own.
Now, who would replace Goebbels depends on Hitler, I would wager that someone within the Ministry, such as Werner Naumann, would have the spot. As for Gauleiter of Berlin, that would be a position Göring and Hess/Bormann would heavily dispute as the control of the capital is crucial in the power plays of the party.
Naumann didn't become State Secretary in the Propaganda Ministry until 1944. In 1938 it was still Karl Hanke, the future Gauleiter of Breslau, who fell out of Goebbels' favour because he took Magda's side and supposedly had an affair with her. He was succeeded by Leopold Gutterer in 1941. Given his conduct as Gauleiter, he'd be no less fanatical than Goebbels (though Hanke tried to run away after turning Breslau into a pile of rubble). Naumann was already working for Goebbels in 1938 as a troubleshooter of sorts, but his position was only that of a personal Referent, basically a personal aide.
It is worth noting that Goebbels, contrary to popular belief, wasn't the almighty dictator over all German media he liked to present himself as. Instead he had to share that position with Otto Dietrich, the Reich Press Chief for both the Reich Government and the NSDAP, and Max Amann, the President of the Reich Press Chamber and Reichsleiter for the Press who controlled the Party's publishing empire. Being a Reichsleiter theoretically made Amann Goebbels' equal in the Party hierarchy, though as President of the Reich Press Chamber he was theoretically also his subordinate. For example, the Völkischer Beobachter, the most important Nazi newspaper, was part of Amann's bailiwick. Since Amann ran the Eher-Verlag, which was the central publishing house for the Nazis, he was also the main publisher of Goebbels' books. Thus both Reichsleiters were dependent on each other.
Goebbels' domain was the radio, but most people still got their news from newspapers. Theoretically, Dietrich was Goebbels' subordinate as a State Secretary in the Ministry and Vice President of the Reich Press Chamber (which came under the umbrella of the Reich Chamber of Culture, which was run by Goebbels), but at the same time he was his theoretical equal in the Party since both were Reichsleiters. If this all sounds messy and contradictory, that's because it is. Goebbels diaries show that they feuded a lot. Dietrich was quite influential when it came to the press in the early years of the war, since he was able to issue directive about how the press was supposed to report things from Führer Headuqarters. Dietrich was a regular member of Hitler's personal entourage and also attained the rank of SS-Obergruppenführer. Despite Goebbels' objections, Dietrich was able to appoint his friend Helmut Sündermann as deputy press chief of the Reich Government. He unsuccessfully schemed to create an independent ministry of the press under his leadership. Goebbels finally managed to persuade Hitler to sack Dietrich in March 1945, but by then it didn't matter.
So with that in mind both Dietrich and Amann would be strong contenders, in my view. Of the two Dietrich seems the most ambitious. I could see Hanke getting appointed to run the ministry instead, but his position would start out quite weak since he'd lack Goebbels' positions in the Party. It is worth noting that Goebbels was also Reich Propaganda Leader of the Party (that's why he was a Reichsleiter), but the Party Reich propaganda office had languished in irrelevance since his appointment as propaganda minister. So I could see the Control Faction (Hess/Bormann and their people in the StdF) try to gobble it up and revitalise it.
As for Hess, if he gets the job I don't think that would stop his flight to Britain. As newer research shows, he didn't do it because he was sulky about being marginalised (which he wasn't, the StdF remained a very relevant institution in both Party and State), but because he feared a two-front war since he knew that the German invasion of the Soviet Union was imminent, and Britain had yet to be finished off. The original plan was to arrange a meeting between Albrecht Haushofer and the Duke of Hamilton, except the letter to the latter was intercepted by the Home Office. Only when that didn't work out did he decide to go on his delusional quest.