Is there any way you could find a different camp? Most of the ones around my area are at least six days long. There are some available for more advanced riders, too.
I will admit that consistent lessons are better for learning how to ride.
However, if you really want to learn things about basic horse care, a horse camp is probably one of the best ways to go.
If you have a choice, a camp where you go to a place and sleep over there would probably be your best bet.
I have very fond memories of horse camp that I will probably never forget. It's simply a blast, even if you're advanced compared to other people. That's how my cousin and I were at last year's horse camp. There's something to be said about being able to help people and having fun relaxing!
If you go alone, you'll meet new people. I'm very shy too, and the first time I was away from home for more than three days was at my first sleep-away horse camp. I hung out with the group I was with and had fun!
I totally understand about your mom. It's been an adventure trying to get my parents to let me ride. Eventually, little by little, they let up! Tell your mom that a weekly commitment to the barn doesn't take a huge amount of time, unless your driving an hour to get there or something. Tell her that it is very possible for children to be at the barn even if the family isn't horsey. It just takes a little time and effort from either her or your Dad to drive you, which I don't think is too much to ask.