This is a point exactly parallel to my last point about promising. It is widely thought that x can only sincerely say that p if x believes that p. That claim is subject to a simple counterexample. An expert entomologist, German speaker and friend of mine gives me a sentence s of German, and he assures me that the proposition p that s expresses (a) is about insects, (b) is true, and (c) is not believed by me. I know no German. I then utter s to some German speakers. In so doing, it seems, I say that p, though I do not believe p. But I am sincere—at least, I violate no requirements of integrity in speech, of which sincerity is one.
Maybe this example does not impress. Perhaps you're worried that by uttering s I do not manage to say that p, because speaker meaning is essential for truthtelling. Or perhaps you modify the sincerity condition to say that one only sincerely utters s if one either believes the proposition expressed by s or believes that s is true. So let's move on to a second counterexample.
The following is sufficient for sincerely saying that p: one knows that were one to say that p, then p would be true. Antecedent belief that p is not required for sincerely saying that p. And consequent belief that p is not required, either, because sincerity in speech is not affected by what happens after one has spoken—and we can imagine that one is killed right after saying p. Here is a fun case. George, whom I trust, has promised me that he'll dance a jig whenever I tell anybody in his presence that he'll dance a jig. So I say to you: "George will dance a jig." I need not have any antecedent belief that George will dance a jig, because unless I succeed in saying the sentence, he probably won't dance it, and I am not sure I'll manage to say the sentence, because I have lately had trouble enunciating the word "jig". Nonetheless, I do not offend against the aspect of the virtue of integrity that has to do with sincerity.
It likewise follows that sincere assertion of p need not be the expression of a belief that p. One might think that perhaps it is the expression of a conditional belief: if I were to make this assertion, then p would hold. But it is very implausible to suppose that our assertions express this belief. Surely when I say that 2+2=4, I am not speaking about speakings.
So, it does not appear that sincere assertion need be the expression of any belief at all. However, a sincere assertion presupposes a belief, though not necessarily a belief that is expressed in the assertion.
The above leaves open the question whether believing that p, while not necessary for sincerely saying that p, might not be sufficient. Here, I do not know. The following case is one to think about. I falsely believe that I cannot utter the word "frog" due to a deep trauma. Let s be the sentence "I cannot utter the word 'frog'" (I count uttering "'frog'" as an uttering of "frog"—make this stipulative). I consider uttering s. I believe I can't succeed. But I can still try. And so I try, and succeed. Was I sincere? I am not sure. I knew that I was trying to say something which, if I succeeded, would be false. However, it is also not obvious to me that I am insincere. It's a tough question.