When you say "it sounds correct", you are speaking about what you expect about "it".
When you say "it sounds correctly", you are speaking about the way it sounds, for example, "That note sounded correctly"
However, the last sentence is almost not used because of common use which becomes grammar later.
For example, in Portuguese, we can say "He died happy" as well as "He died happily" but this is not allowed in English, not because of logic but because of common use.
You must say "He died happily". However, you can say in English "He died bored"... As you see common use overcomes logic.