The default position
Before we start addressing your doubts head-on, we should think about what gets to be the default view. That is: while we're waiting for the doubts to be resolved, should we abandon moral claim-making altogether? Or should we assume that it's okay to make moral claims until we're shown otherwise?
On this page, I'm going to suggest that it's okay to carry on making the moral claims until we're given good reason to think we shouldn't. Here's why.
As I mentioned in the introduction, it's pretty standard for all of us to go around making moral claims. We're always talking about what people may, may not, must, mustn't, should, and shouldn't do, morally speaking, and about what kind of behaviour is good, bad, better, worse, best, or worst. You'll have defended or condemned all sorts of behaviour—both yours and that of others.
It looks, then, as if you and everyone you know have been acting all this time as if there is nothing particularly contentious about making moral claims. You do it all the time, and you appear to take the claims at face value. What I mean by that is that you treat the claims as perfectly capable of being either true or false, independently of what the person making the claim thinks.
In light of this, it seems to me (and many others) that the default position has to be that moral claims, understood as capable of being true or false independently of the view of the people making them, aren't especially problematic. We make them and argue about them all the time, just as we make and argue about claims regarding causes, statistics, the weather, efficiency, football, people's feelings, and any number of other things.
No one's saying that all the claims we make about these things are uncontroversial. But they do look as if they can be true or false independently of the views of the people making them. And just from the way we talk about them you wouldn't think there was any reason to pick out moral claims as especially problematic in that regard.
If this is all about right, then it seems that you do need some kind of reason to be suspicious of moral claims. It's certainly not the default position that moral claims are fishy. So, unless we're given a particular reason to start treating them differently, the presumption has to be that there's nothing wrong with them.