Hi John,
Mark is correct, you just have it installed in the wrong directory. You could move it just by copying the entire contents of that folder to the public_html folder and that will work. But it sounds like you've got some complications from deleting previous versions, so it might be better if you back up a step.
You won't be able to do a clean install to your home directory because you still have the drupal sites folder there. This is really perplexing to new users because you click delete and it won't go away.
Here are some directions for getting rid of that folder
www.ostraining.com/blog/drupal/how-to-delete-the-drupal-sites/ If that's more than you need to know. Just ask your host to delete the sites folder from the home directory. You could leave the install in the times directory if you want to use as a test site. But if you go to delete it, you'll have to delete the sites folder there as well.
Now reinstall drupal, but this time, watch the whichever one-click installer you use. While you're going through the prompts, it will ask you which folder you want to install drupal, and it will probably be automatically filled in for you with:
www.sportingtimesmagazine.com/drupal.
When you did the first instally you most likely replaced drupal with times. This time, delete drupal and leave that part blank so that it only says
www.sportingtimesmagazine.com/ That will give you a clean insallation of drupal in your home directory and it will work the way you want. If you delete it again, and want to start over, just remember to delete the sites file manually or with the help of your hosting company.
Let us know if that helps you out.
Cheers,
Ed
PS. The more trouble you have now, the easier it is in the future. Keep plugging at it! And keep asking questions.