Hmm. Well, I don't see any "objective" morality. So in that sense, sort of yes; there isn't any objective or absolute "right or wrong," so I don't concern myself with objective or absolute right or wrong.
However, as far as what's best for humans and our planet, or myself and my family, there *are* clear things that are smart and reasonable ("right"), and stupid and unreasonable ("wrong"). So in that sense, I do concern myself.
Amoral would generally mean 'without moral.' Immoral would mean 'not moral.' They're not synonymous. Immorality requires one to define morality first. In the LDS context, drinking coffee is immora; in the Muslim or Jewish context, eating bacon is immoral. Outside of religion, the same arguments can be made. For a vegan, eating meat products may be considered immoral. And so on.
As far as OP's question goes, I occasionally make amoral decisions. As an example, I once took a contract to provide services for the county chapter of the opposition political party of the one I belonged to. Assisting them might be considered 'amoral,' but such is the way the cookie crumbles. Businesses will often lay people off -- such decisions will often happen outside of a moral context, as business is concerned with ethics and legalities, not morality.