Drying leaves and black branches are not a good sign - your fig tree, as you suspect, may possibly be dying. Before we jump to that prognosis, however, let's try a few things. I would like you to give your fig tree a scratch test on the trunk and look for green living tissue- for information on how to conduct a scratch test, please visit the following link:
In addition, try snapping the end of a black branch toward the tip. If it snaps easily and is brittle then cut the branch back towards the trunk and see if you see any green inside the branch.
Remember, green is good and what we are hoping for. If you see signs of life, the best thing you can do is follow the care plan of a fig tree (see link below) and keep monitoring it for growth.