Sincerity and truth are not related.
If I believe the Earth is flat, I can defend it with all sincerity and authenticity. And it is still not true.
Also, one can be false and evil only telling truths ("How am I then a villain to counsel Cassio to this parallel course, directly to his good?")
Truth is reality. We live in truth. The obligation to say the truth is not related to moralistic demands but to be able to stand in soul and action in relation to things that are real and not illusion. We have to tell the truth to ourselves, about ourselves and the world. That is character and the beginning of the purification of the nous.
Your example is good, but sincerity and truth are sometimes intertwined. Sincerely relating the fact that an accident occurred one hour ago on the busy intersection of X and Y and recommending an alternative route that takes you 15 minutes out of your way, is a truth, but perhaps not at the moment it was spoken (it might or might not). Your Shakespeare quote, is not within my knowledge. It sounds like an incomplete truth from the perspective of the one saying it. I do not think I am required to know this quote.