However, "President Dunn sent a letter to myself" may sound equally acceptable to "President Dunn sent a letter to me." Depending on what you are accustomed to, "myself" might even sound better, but it's wrong.
While "myself" and "me" are both objects, "myself" is what is called a special object. You should use "myself" and not "me" as the object, only when you are the subject of the sentence. Example: I could not dress myself.
The reflexive personal pronoun myself gets tricky but just remember that myself is always used as the object of a sentence and/or as an intensive pronoun to add intensity to a sentence. Myself is never used as a subject pronoun.
Notice, though, that in the first example, the subject (Barry Manilow) and object (me) are different. Thus, the object pronoun is correct. In the second example, the subject (I) and object (myself) are the same.