Do you have a link to the pattern or information about where to find the pattern? That might help.
Did your pattern tell you to use 10" needles? If so and there is nothing about a longer needle maybe it is constructed differently that you think. You usually don't make a whole back piece and then add on sleeves.
Maybe you knit the sleeves totally separate and sew them on.? But that doesn't explain what you said here:
It does say that after casting on the sleeves I will have a lot more stitches then when I first started the back part of the sweater.
Maybe you don't have the whole back done but only up to the armholes. Some sweaters are made that way, make the back to the armholes and then cast on stitches on each side for the sleeves. But then you would probably need a longer needle. Circular like Marilyn said.