Upgrade Civi first, or Drupal first?
Kotodharma,

Welcome to stack exchange. Its a huge pain to upgrade CiviCRM from 4.4 to 5.6. If your site has lots of customization then the upgrade might tend to break at some point if you're not lucky enough. Upgrading Drupal or CiviCRM first doesn't matter. But yes while upgrading CiviCRM I would suggest you to perform incremental upgrade so that if anything breaks you don't need to start it all over again. Here is what i would suggest 


Upgrade to last version of 4.5
Upgrade to last version of 4.6
Upgrade to last version of 4.7
Upgrade to 5.6.0



  Note: Take backup at every stage so that you can revert it back if upgrade fails.


HTH

Pradeep