I do not think he has broken his back. I think he has metabolic bone disease, or at least hypocalcemia.
How old is your dragon, you said around 2-3 months old? At that age, they need calcium 5 times per week & UVB lighting. The All Living Things is not a UVB light, it is just a UVA light, for a fish aquarium I believe.
You will need to get a Reptisun 10 flourescent tube bulb, & you can order that from http://www.petmountain.com
It will be much cheaper online than from a petstore.
What type of basking light are you using? What do you use to measure your temps with, a stick on thermometer, a digital probe or a temp gun?
Good he is on carpet right now, please keep him on that & do not switch to sand until he is an adult.
I would use fresh food not the canned ones. They are much more nutritious than the canned foods.
Do not use mealworms as a staple, they are too fatty. Instead use appropriately sized crickets, & maybe some phoenix worms for variety.
Please let us know how he is doing.