Are you stating that your J1 expires at the same time that your classes start? Should we assume that you have no home country residency requirement on your J1?

If you're telling us that you can stay without going out of status, you will be in J1 status until you go into F1 status. If you are telling us that you are going to go out of status in between the time your J1 status ends and your F1 status starts, you'll be out of status unless you leave the country before your J1 status ends.