politeness - How should I correct my spoken error?

Being an English language learner for about 30 years, I still make mistakes.
In a written language, when I have misspelled and noticed it in a timely manner, I simply navigate to a wrong word and edit it.

However, in a spoken conversation, when I use a wrong word, I must stop and correct myself by saying a correct one. I've seen many ways to do that, but I'm still not sure which one to use:

He's going to school...

• ...er... she's going;


• ...sorry, I mean, she's going;


• ...oops, she's going;


• ...I'm sorry, let me restate it from the beginning, she's going;

What is the grammatical and polite way to correct myself in a spoken conversation?
Let's stick to a formal/business context if it matters.

Also, what part of a sentence should I repeat to make it clear?

Answer

They're all fine, as is simply restating without comment:

He’s going to school— she’s going to school at Princeton.

On this site it may be worth observing that nobody expects spoken English to be perfectly expressed, from native speakers or anybody else. Speech is improvised: people start a sentence with only a general idea of where it’s going, change their minds in the middle, have second thoughts, find new ideas or expressions suddenly occurring. So if you listen closely to conversations, or if you read a literal transcript, you will find that errors are abundant. On the page, excited speech may be almost incomprehensible.

Consequently, most errors in speech are ignored, by both speakers and hearers. It’s taken for granted that errors will occur, and for the most part hearers make the necessary corrections automatically, from context.

Back when I was active in theatre I always told my actors to play through errors as if they hadn’t happened—“The audience’ll never know you screwed up unless you tell them.” The same principle operates in ordinary conversation: don’t make a correction at all unless there’s a real danger that what you said will be misunderstood. In your example, for instance, there's no need to back up unless two people, one male and one female, are under discussion.