It is tempting to define promising in a way that centrally involves some condition like "S intends that the utterance of T will place him under an obligation to do A", as Searle does. Of course, there will be other conditions, but this is the one that most clearly distinguishes promises from other illocutory acts.
However, there are many ways that an utterance that expresses that one will do A could place one under an obligation to do A, without the utterance having the illocutory force of a promise. For instance, when I assert: "I will not deny having made this assertion", I have thereby put myself under an obligation not to deny having made that assertion, not by making a promise (I was asserting, not promising, hence my use of "will" in place of "shall"), but simply because it's wrong for me to lie. Searle's other conditions will rule this out as a counterexample, but it is easy to imagine a social institution like promising but whose moral oomph is not that of the promise, but of something else, like obedience to an order.
For instance, Fred might command me that whenever I say: "Pursuant to Fred's command, p", I must bring it about that p. (I think that I can make a case like this fit all the other Searlean conditions.) If Fred is a legitimate authority, then by making an utterance like that, I gain an obligation to bring it about that p, but I do not gain promissory obligation but an obediential one. An easy way to see that the obligation is not promissorily grounded is to note that if I say: "Pursuant to Fred's command, I will take you out to lunch", the obligation is not to my interlocutor, as it would be in the case of a promise, but to Fred. Of course, one might modify the central condition by saying that the utterance will place S under an obligation to the interlocutor. But, nonetheless, I can suppose that I say "Pursuant to Fred's command, I will take you out to lunch" to Fred. In that case, I may owe the obligation to Fred, if Fred is the sort of authority obedience to whom is owed to him (God is that sort of authority), but it is still an obediential and not a promissory obligation.
I wonder whether promising can be defined in a vocabulary that uses normative and even moral vocabulary but that does not distinguish promissory normativity from other normativity.
This is a general worry about "normative" accounts of various phenomena. Consider, for instance, some normative account of assertion, where a central condition for T to be an assertion that p is that T is only permissiible if the speaker is justified in believing that p. Unless one specifies the kind of permissibility at issue, we open ourselves up to weird counterexamples, such as when a superior commands us to write down a list of yes/no questions to which one justifiably believes the answer is affirmative—it is, then, permitted to write down the question whether p only if one is justified in believing p. Here, the normativity is of the wrong sort, and a correct definition of assertion needs to specify what the relevant kind of normativity is. And I don't know that one can do so without saying that we're dealing with assertoric normativity.
(In the end, I think all normativity is moral. But there are different kinds of moral normativity, as there are different kinds of moral reasons.)