I wonder what else (that sounds more natural to a native ear) I can say about being sorry for repeating myself. Can anyone help?
Your statement ("I'm sorry for repeating myself") sounds perfectly natural to my ear. Some other options:
Abby's answer has five good options. If you want to add some idiomatic expressions to your list, you could add:
The expression beating a dead horse would be appropriate if you think the other person might be getting a little impatient with the fact that you've been saying something over and over again. One website says:
On the other hand, one can apologize for sounding like a broken record without any impending exasperation required. From UE:
I might use the broken record expression to describe a small child who is saying the same thing over and over and over again. I might use the dead horse expression to describe someone at work who feels adamant about something – who, despite not being able to persuade anyone that his argument is valid, keeps bringing up the matter over and over again. The expressions are a bit more specialized in meaning, but they could still both be used while someone is apologizing for repeating themselves, depending on the situation.