I've never heard conservatives say that rejecting their values is a rejection of all values. I don't think that a conservative would say that liberals have no values; liberals have very clear values. I can't speak for the ones you've interacted with, but I've never heard that.
Regarding moral relativism, the argument, as I understand it, is that post-modernism rejects any meta-narrative; you decide for yourself what the narrative is. Moral absolutism requires a meta-narrative of some form. This is right because God told us it is, or because this is the core value our nation is founded on, etc. So without a meta-narrative, you have to define your own, and so you are left with moral relativism.
It's possible that post-modernism is no longer widely held. I'd probably be the last to know about it. If that's the case, then maybe there is moral absolutism. Certainly there seems to be an idea in some circles that protecting the environment, and/or everyone has a right to express their sexuality however they want to are absolutes. But a truly moral absolutism provides an absolute basis, and I'm not aware of any absolutes for these. Environmentalism is a pragmatic source: if we don't do it, we might die off, but perhaps dying off is actually best. (I don't agree, but philosophically speaking) And what basis is there for everyone having a right to doing things? There are things we decided we don't have the right to do (kill people, for example). Why, exactly, does everyone have the right to express their sexuality however they want? Moral absolutism requires some fundamental, unalterable reason. Moral relativism simply requires "I think this way."
I'm not very sure what most people's world views are, but everything I am aware of points to a moral relativistic view, rather than a moral absolutist view.