What does “sign the final flourish” mean?
Solution 1:
When you sign something with a flourish you make a kind of bold dramatic motion of the pen.
I think it's a metaphor to living the end of his life, and how he would like to deal with it, in style and displaying good character.
Solution 2:
Generally it just means to give things closure in an elegant way, similar to “dotting your Is and crossing your Ts,” but perhaps with more grace. The connection here to the end of a life adds poignance to it.