I would just like to add some additional and personal insights based on experience, and having three sons of my own who are all expert skiers.
If, as a family you can go to a resort for a week and become immersed in the sport, with your sons going to the all day academy, ski school, kids groups, or whatever the resort chooses to call it. Anyway, put them in that all day program, lunch included. They will be skiiing with their peers, and will make much greater progress than skiing with you, and your wife. You and your wife do the lesson thing with other adults. After you get your sons from their ski school [usually about 3-4pm.,] then as a family take a few runs together. You will be amazed at how much fun you will have, and how quickly and how much progress your sons will make. This will be true quality family time spent together doing a sport that can last most of a lifetime.
Now is the time to do this, because it is a great family sport, but soon your sons will both be in high school and it will be very difficult to get your sons involved in another sport. Involving them now means they may want even want to take ski trip vacations with you once they are in High School and even College.