Coming out on the Other Side.
If this were a therapy session, we would spend as much time as needed to sit in the context of your current state. There is much to unpack with what you have chosen to write. Mom's terminal cancer, depending on your relationship, could be many things to try and understand. We would try to determine which areas of her cancer and the possible outcome that is most pertinent could be a necessary beginning. Nan dying, is another thing to unpack based on your relationship with her and how this has affected you. Your cat, the reality of pets and their brevity in our lives, how innocent they are, and how quickly they seem to be taken from us. Not to mention, there is still work, bills, relationships, possible social interactions, and everyday stressors. Trying to "make sense of things" is often one area to start working on.
Stop trying to make sense of anything. You didn't say you were, but it's a typical occurrence that people get stuck in this way of thinking where they are trying to figure something out, even if it is how to move on, and it prevents them from living in the moment. Even with unfavorable moments, they are your life, and you must deal with them on life's terms. Death and death scares are often so abrupt they shake us. The death of someone close is like a mirror showing us our brief life and mortality. No matter what you do in this life, you too will be like nan, mom, and your cat. Don't try to make sense of it, legitimize it, or feel better. Just be in the moment. This is your life right now, it has never offered a more opportune time to be a part of it.
Experience the moment, but do not indulge in grief. Let yourself be sad when you are sad, happy when you are happy and relieved when you are relieved. Do not indulge in whatever thoughts that come up. At this point, you cannot afford to. There is already too much taken from your emotional bank to try and give it to another cause. Just sit with what shows up, remove expectations as to what to do with it, and try to reduce life stressors because they still exist.
Now is not the time to make big life decisions. Now is the time for rest and learning what pain is, what is really going on, and how to deal with it. As the title of this page indicates, you will come out on the other end stronger. Through tough times, you force your mind to react to painful, unavoidable situations where one builds their best self, the true and vulnerable self. Let this happen. Carve out time in the day to connect with your physical and spiritual self and feel the emotions you may have thought were inconvenient and wanted to rid yourself of. Do not try to do what you think you SHOULD do, but grieve when it comes up and talk to people about it.
I am not sure of the status of your spiritual life, but death often reveals to us that we are not meant to be here very long (I think I have said that a few times now). Accepting that this life is short, you can now start to live for what matters and how you perceive your afterlife. For Atheists, it can be sad to think there is nothing. To Christians, it is a joy to pass into the next, eternal and perfect life. Other religious groups have their own outlook that should shape how we approach death and life.
Let go of trying to find the energy to recover; that will come when it needs to. Think about if your body gave you that energy too soon. You would go about your life and not take this time to do what is necessary to learn about what grieving means to you and who you are when you are the most vulnerable and probably the most important thing, what is important in life.
You are doing great in thinking about the things you asked about, and I encourage you to think of what can help you and what you would need or want to learn about yourself, others, and relationships going forward.