There is some truth to your complaint, though I would suggest that the Declaration's "we hold these truths to be self-evident" is the part more properly identified as being "held up as axiomatic." On what other basis, after all, can the rights life, liberty and pursuit of happiness properly be labled as "unalienable?" They are axiomatic in the sense that they are "endowed by our Creator." I think the left doesn't actually believe this to be true -- at least, not in the way the Founders did, and also not like at least some conservatives do.
In that case, of course, the objective basis for the correctness of the Declaration would depend on a "first reality" that actually includes a Creator.
I like the Adams quote not for any "axiomatic" reasons, but rather for its practical (if unspecific) statement of the necessary conditions for limited government. Absent the self-policing nature of a "moral and religious people," it is difficult to imagine a system of "limited government" as not descending into anarchy; or, in order to prevent anarchy, the government would tend toward tyranny.
The question for you is, is there anything that you could offer as a "basis for conservatism" that is not in some sense "axiomatic" in a "second reality" sense?
Probably not. But with regard to issues like constitutionism and the intent of the Founders, there is a considerable body of work available for consideration. I'm skeptical of conclusions drawn from one quote by one man, and a single sentence lifted from the Preamble.
Axioms can properly be constucted an applied in mathematics. The term is something of a misnomer when applied to political philosophy, but generally speaking the less you base your conclusions on the more "axiomatic" it's going to appear, and I think there's a lot more to the situation than what's being presented as the relevant facts.