Making a moral claim means, more or less: "I am right and you are wrong and you should do what I say"
Well your attempt to explain what a normative claim is actually includes a normative claim so I don't think you've successfully dissolved the question. You are "right" about what? Facts? The world? What kind of facts? What kind of evidence can you offer to demonstrate that you are right and I am wrong?
"what should happen when folks make incompatible moral claims, or claim incompatible rights"
That "should" is there again.
When we do ethics, we abandon what might be called (in a perhaps naïve and philosophically incorrect sense) "moral absolutism" or the simple practice of just making moral claims, and start debating them in public.
I don't imagine there ever was a "simple practice of just making moral claims". Moral claims are generally claims made on others and they are speech acts which means they exist to communicate something. People don't spend a lot of time making moral claims that everyone agrees with and abides by which means it's pretty much in the nature of a moral claim to be part of a debate or discussion.
I can't see the importance or the force of the distinction you are trying to make.
What kind of evidence can you offer to demonstrate that you are right and I am wrong?
Who says I need "evidence" to argue that you should do something? I could rely on my perceived authority - in fact, you could take this as a definition of what "moral authority" is all about. Sometimes that moral authority comes from religion (or cosmology, more generally), sometimes it's derived from tradition, etc. So I have to dispute your claim that:
it's pretty much in the nature of a moral claim to be part of a debate or discussion.
since ...
I think there’s a confusion in our discussions of deontology and consequentialism. I’m writing this post to try to clear up that confusion. First let me say that this post is not about any territorial facts. The issue here is how we use the philosophical terms of art ‘consequentialism’ and ‘deontology’.
The confusion is often stated thusly: “deontological theories are full of injunctions like ‘do not kill’, but they generally provide no (or no interesting) explanations for these injunctions.” There is of course an equivalently confused, though much less common, complaint about consequentialism.
This is confused because the term ‘deontology’ in philosophical jargon picks out a normative ethical theory, while the question ‘how do we know that it is wrong to kill?’ is not a normative but a meta-ethical question. Similarly, consequentialism contains in itself no explanation for why pleasure or utility are morally good, or why consequences should matter to morality at all. Nor does consequentialism/deontology make any claims about how we know moral facts (if there are any). That is also a meta-ethical question.
Some consequentialists and deontologists are also moral realists. Some are not. Some believe in divine commands, some are hedonists. Consequentialists and deontologists in practice always also subscribe to some meta-ethical theory which purports to explain the value of consequences or the source of injunctions. But consequentialism and deontology as such do not. In order to avoid strawmaning either the consequentialist or the deontologist, it’s important to either discuss the comprehensive views of particular ethicists, or to carefully leave aside meta-ethical issues.
This Stanford Encyclopedia of Philosophy article provides a helpful overview of the issues in the consequentialist-deontologist debate, and is careful to distinguish between ethical and meta-ethical concerns.
SEP article on Deontology