How do I move on?
There is nothing that can be said here to make anything better. What happens in life are these inevitable situations where pain stacks up and pushes us to our limit. You were not yet an adult, but your parents' children, and now you are trying to visualize life without them, meanwhile living with the pain of watching them both die so abruptly.
Everybody will say this, but what would your parents want for your life? After all, they were the ones who sacrificed to get you to your adulthood. If they committed to their purpose, it was to get you fashioned into a responsible adult who, if blessed, takes on a family and does what is good for others. You are seeing this situation through the lens you are familiar with in your life. There is an alternative view: practice empathy and see the same situation from your parents' eyes.
Since you are a parent, you know what it is like to watch a cute little, helpless baby start to roll over, crawl, then walk, form a personality, and before you know it, start to talk back. The psychological development of children is just as much a milestone as walking or talking.
Please stick with me here.
However, when our little child minds become aware of others, such as our parents, we become aware of ourselves. When we become aware of ourselves, we start to say and do the things that expand on this "self" idea. Parents have to watch it go down every time we say "no," or "I hate you," or worse. You see, your parents have already sacrificed to do what was necessary, not for the pain itself, but for the joy of seeing you grow up and do the same for others.
Love doesn't stop at death, but continues on in teaching and modeling. Your parents gave you love, and you innately know love. Because of their sacrifice and their gift to you, in raising you and watching you grow and keeping you safe to grow into the person you need to be for others, they served their purpose.
I guess this always comes down to purpose. What is it in life that is worth all of this? What do we think of our life? Is it for comfort, or for purpose? Often when we mourn and lose hope in any purpose in life, it's because we seek the familiar comforts. Your parents served their purpose so that you could be comfortable? No, they sacrificed, innately, to provide for you to become something for others as well. This is the basic premise of humanity and one that you are responsible for, from them.
Use the gift they gave you, their time and genetics, to go out into the world and do what you are supposed to do. If, at this point, you do not know what that means or what that looks like, find what matters most to you. Whether that is service to others, or maybe it is an encouraging word. Whatever your view is on the afterlife, some people become more aware of their transcendent, spiritual self. Your job now is to appreciate mom and dad through living the life they would have wanted for you.
Your children do get to meet your mom and dad, their best parts, in fact, through you. I hate sounding wishy-washy, but this is reality; your parents are not "gone," but only in one form. Your parents invested their entire life into you, and now your kids get to see what that is like as you filter out and give to them the best parts. You have a job, and though sadness is present, you have things to do.
Bring the sadness with you. Bring the pain and hurt with you on this. That feeling right now, what you are experiencing, hold on to that. Not as a victim, but remember it because one day, you will need to tap into that pain to relate to someone else and what they are going through. No, your situation will not be theirs, but, you will know what it is like to hurt. Maybe that will be for your own kids, as they grow up and face this harsh world. Maybe one day, you will sit at the foot of the bed while they are crying because a friend was mean or maybe they themselves messed up. Your job at that moment will depend on you today and what you contribute to yourself in finding hope in the darkest places. You can actually find gratitude in all of this when you think about it.
Your parents did all of this, to give to you, your kids, and the world you experience the gift of a fully formed and caring you. Show the world what they sacrificed for.