Hello and good for you to have a goal of using your solo time to improve.
Pros and cons to privates versus group lessons. If cost is not a concern, I would ask for a minimum of a level two, patient instructor and do private half days. I agree with the statements above that it is hard to concentrate and actually keep improving in an all day lesson.
Groups can be more fun if you end up in one with similar skiers, or even different ones where you learn how we all struggle with different skills and emotions on snow. They can also be challenging if the fit is poor amongst participants and your instructor not skilled enough to manage the differences.
In a private you could specifically work on your personal goals, which seem to be going faster. I often help guests with that, but usually focus on mastering going slow and stopping on a dime before doing some straightlining drills with them.
If you end up with an instructor you do not like, it is absolutely OK to ask for a different person. You might even say that right away, that you really like different ways of learning the same idea, and then you are less likely to feel awkward if you ask for a fresh face the next day.
Some areas you will need to prebook as instructors are quickly filling their calendars with lessons for the whole season.
Remember that you will likely have big speed demons for sons as they become better and even faster skiers. I suggest learning to love the sport for your own pleasure. Improvement can and will happen, but speed differences will likely remain....at least that is what I have witnessed. We just don't get that fast when we start as adult and wise women (wink wink).
On a different note, I don't want to assume anything, but my guess is that your sons are normal boys who are just hard to trick into wanting to turn. I have even had non turning adults, and my 11 year old student was almost taken out by one two days ago! It is often just a phase, and we try to challenge the boys in a safe environment where they do not hurt themselves or others while doing our best to get them to understand the value of turning. They often figure it out very fast once we bring them to a steep run, but beg to return to the more playful greens until they are motivated to make the step up to turning and steeps.
Again, I would encourage you and your family to make peace with the differing skill and speed levels, or all will remain frustrated. The key is the find the joy in the level we are at on any given day in our love of skiing. I much prefer being around an advanced beginner who made progress than an expert skier who is bored and complaining about the slowpokes around them.
Take lessons, don't be afraid to try the many drills your coaches will have you try, and relish in the joy of getting better and faster!