Sincerely is one way of signing off on a letter you’ve written, and it’s also an adverb you’d use when you want to emphasize how much you really mean it. I sincerely hope you understand what I’m talking about.
Sincerity means being completely honest about what you mean, and only those who have sincerity should use the related word sincerely to describe their actions. If you sincerely want to swim in a lake, you’re not at all joking, which explains your bathing suit. You can use sincerely at the end of a note, too: “Dear Mom, I sincerely wish you would stop cutting my sandwiches into heart shapes. Sincerely yours, Kurt.” No doubt, Kurt sincerely means it.