“Really” is my favorite word, so may I share some ideas. Remember: if a language professional says that I am wrong, then you must accept the language professional’s word, not mine.
As the language coach says, your question is not clear. So I shall discuss the adverb “really.” Please remember something very important: the meaning of the sentence depends on WHERE you put the adverb “really.”
(1) Let’s discuss your first sentence.
Mona: You hurt my feelings yesterday.
Tony: I am sorry.
Mona: Are you?
Tony: Yes, I really AM sorry. (Some teachers say that “really” makes the word “am” stronger; some teachers say that “really” makes the words “am sorry” stronger. The sentence means something like: “I very much am sorry.” That is stronger than “I am sorry.”)
Now look at what happens if I move “really”: I am really sorry. Now “really” refers to “sorry,” and the sentence means something like: I am VERY sorry.
Now look: Really, I am sorry./ I am sorry, really. Those two sentences = Believe me. I am telling the truth: I am sorry.
(2) Let’s discuss your second sentence.
Mona: My boyfriend says that he has a new girlfriend.
Martha: Who cares? You will get a new boyfriend in one day!
Mona: Well, I don’t want to lose him.
Martha: You don’t?
Mona: No, I really don’t want to lose him. (Some teachers say that “really” makes the word “don’t” stronger; some teachers say that “really” makes the words “don’t want to lose him” stronger. The meaning is: I very much do not want to lose him.)
Now move “really” to other positions: Really, I don’t want to lose him./ I don’t want to lose him, really. Those two sentences = Believe me. I’m telling the truth: I do not want to lose him.
And look at this: I don’t really want to lose him. = something like “Well, if he leaves me, I will not be happy, but that’s his decision. I will just wish him good luck, and then I’ll find another guy, I guess. What else can I do?”
(3) Your last sentence is “easy.”
In “You can really sing,” I think that “really” refers to “sing.” If you say “You can sing,” that is nice, but if you add the “really,” you are
strongly complimenting the singer on his/her singing ability: s/he can sing to a very high level of excellence.
If you move “really,” as in “You really can sing,” some teachers would say that you are making the word “can” stronger; some teachers would say that you are making the whole idea “can sing” stronger. The sentence means something like: Wow! You very much have a talent for singing.
Finally, look at: Really, you can sing./ You can sing, really.
Mona: Mr. Smith told me not to sing in this room.
Joe: Don’t pay attention to him.
Mona: But he told me not to sing.
Joe: Really (believe me), you can sing. = I am telling you the truth: you have PERMISSION to sing.
Mona: I had a very bad accident that damaged my throat. I can’t sing anymore. I am so sad.
Joe: That’s not true. You can sing. (= you have the ABILITY to sing)
Mona: I do not believe you.
Joe: Really, you can sing. = Believe me. I am telling you the truth: the doctors tell me that you still have the ability to use your voice for singing. Just start singing. You will discover that I am telling you the truth.
I think that “really” basically means three things:
(1) “Very” = You are really intelligent.
(2) “Very much” = I really appreciate your help.
(3) “Believe me” = Really, I did not do it!