There are three things that may happen,
One is well everything will die
Second is that Humans may find ways to survive through scientific means or migrate to other planets which will be quite possible if you ask me..
Third is that Humans may evolve to adapt or a new spices that are much like Humans but better will replace human kind...
And regarding the Sun's health i.e well if it dies idk what gonna happen cause the Universe is Mysterious and kind of scary if you believe in aliens