First of all. If you just submitted you need to wait for FAFSA to send you the "processed" email, which takes up to two weeks. After that, yes you would call each school after giving them a few weeks to see if they have received and downloaded it before you delete them.
You could also call each school and give them your individual code at any point so they can manually request your file when they are ready for it. In this case, you wouldn't have to delete anything, they would be doing the leg work. Some schools may tell you they won't do it that way. So YMMV with each school.
Sent from my iPhone using
SDN mobile