This is a major issue for many families. And I am right in there with you at marveling how families coordinate home school and other interests without becoming overwhelmed. I was never able to do that myself with only three children and few assorted pets. I used to wish that I could bi-locate, but we must look for a more natural means of coordination.
We handled outside interests by keeping the children together as much as possible, and we didn't allow outside interests until the children were older. So everyone ran in the children's races, and everyone was involved in acting camps and classes. Everyone took confirmation classes at the same time. I know other families who used the same method. It seems that that is what you are doing with the violin lessons.
Another possibility is that each child has one outside interest. That means the child must prioritize and choose one activity over another. Yet another idea would be to coordinate these outside interests with other parents. Perhaps you can take turns with other parents driving your son and other boys to Scouts meetings and your daughter and other girls to gymnastics workouts.
If you are not already doing so, you may want to use your drive time and other down-times for lessons. We would frequently play word games and listen to educational tapes while driving to and from Mass, the library, races, the theater, the orthodontist, and visits to family. My children would sometimes do their schoolwork in medical waiting rooms.
Your home schooling life is going to be busy and overwhelming at times, and you aren't always going to be able to make nutritious meals and keep a tidy house. That is okay. Try not to judge how you are feeling and how your schedule is going or not going by what other families are able to do or seem to be able to do. (One busy mother with two children who appeared to have the perfect house admitted to me that she didn't cook, and I later discovered that she had a network of friends and family whom she relied heavily upon.) Your energy level and your family's needs are different from other mothers' levels of energy and other families' needs. Focus on prioritizing and doing what is best for your family.
Your emotions are going to affect the rest of the family, as you probably already know. Therefore, take time for yourself to be peaceful and quiet so that you can give that gift of peace to your family. That may mean that you schedule at least thirty minutes twice a week where you are not available for the children's outside interests. A friend of mine who is home schooling a severely autistic child tells me that her hour in front of the Blessed Sacrament each week is what preserves her peace of mind. She is able to recharge during that time and to realize that she will not be able to do all that she wants for her son and that she is not alone in her struggle.
And you are not alone as you continue meeting your family's needs and wants. I doubt if there is any secret that will work for everyone, but the struggle to maintain that balance is necessary for your family. Your children will learn from your example. May God smile upon you and grant you peace.