The way I see it is this.
Morality is about rules. It is either about following largely static rules, or judging and evaluating people by largely static and unchanging rules. This leads to a lot of unhelpful in-grained attitudes and prejudice. Morality is dehumanising and may actually harm our emotional well-being. Morality is not driven by love or understanding. It "objectifies" people. Morality is about expectations and obligations that are often not realistic.
In religion, obsession over "morality" often leads to fundamentalism, judgmentalism and legalism and is usually the natural result.
Justice is about the actual people, individually speaking. It is about their flaws, their weaknesses, fears, insecurities, personal issues, habits and their personal journey through life.
Everybody wants to be treated with dignity and sometimes the unwritten rules of society that project unrealistic expectations of people lead to them being stigmatised, judged, criticised, condemned and demonised in ways that are unfair, damaging and hurtful.
Everybody has a journey, a story and justice is about living in peace and harmony in our journey through life. Conflicts are points of intersection between two stories and journeys involving conflicting interests, where one inflicts damage on another and/or vice versa.
Injustice is a disruption of the normal, expected progression of a journey through life by another seeking their own needs and interests. It is a destruction of the peace and harmony that existed before, peace and harmony that must be restored by the pursuit of justice. Justice is the restoration of that peace and harmony. It is the process of directing two conflicting parties back onto the journey they were taking before they clashed. The needs and interests of the two parties must be addressed and fulfilled before they can move on. When that happens, the process of justice is complete and a new cycle begins.