There are (at least) two ways of denying moral absolutes. You can say “I don’t believe there are any” or you can say “I believe there are moral absolutes, but (a) there are too many candidates for membership in that category and (b) there is no device, mechanical test, algorithm or knock-down argument for determining which candidates are the true ones.”
What exactly will have changed when one set of philosophical views has been swapped for another? Almost nothing….You won’t say, “Because I believe in moral absolutes, I’ll take this new job or divorce my husband or vote for the Democrat.” Nor will you say, “Because I deny moral absolutes I have no basis for deciding since any decision I make is as good or bad as any other.” What you will say, if only to yourself, is “Given what is at stake, and the likely outcomes of taking this or that action, I think I’ll do this.” Neither “I believe in moral absolutes” nor “I don’t” will be a reason in the course of ordinary, non-philosophical, deliberation.
Now that’s some pragmatism that I can believe in!