I've received instructor permission for a course, but when I attempt to enroll, I receive an error message stating that the class is full even though the class hasn't reached its capacity.

You are receiving an error message because you have a closed lab/discussion section in your cart. To enroll in the course, delete the course from your cart, your instructor's permission still remains valid. Re-add the course to your cart and choose an open lab/discussion section to complete the enrollment process.