The other thing I changed is that the drupal site and the moodle site differed with the absolute URL. The moodle site was loading with the www prefix, but the drupal site didn't have the www prefix. Once I removed the www prefix from the moodle site, the proper moodle session cookie was set and it seems to be working.
If anyone else has a similar problem I have figured out a possible solution. I noticed that the moodle session cookie was still set even after I logged out of moodle. To that end, when a user goes to their account page in our Drupal site I am loading the moodle side in an iframe (0 width, 0 height, no border) in the background so that the moodle session cookie is always going to be set before a user tries to go to moodle.