I wonder what else (that sounds more natural to a native ear) I can say about being sorry for repeating myself. Can anyone help?
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.
Your statement ("I'm sorry for repeating myself") sounds perfectly natural to my ear. Some other options: