I found all the different places that it needs to be updated but wanted to document them here incase anybody else ever runs into a similar issue.
- DeploymentWorkbench > Deployment Shares > Share Name > right-click > Properties > General > Network (UNC) path
- DeploymentWorkbench > Deployment Shares > Share Name > right-click > Properties > Rules > Edit Bootstrap.ini > DeployRoot=\servername\directorylocation
- \servername\deploymentshare$\control > Edit CustomSettings.ini > If the server name is included anywhere in this file, update it
- \servername\deploymentshare$\control\deployment\ > Edit Unattend.xml > Do a find a for the old server name and replace with the new one
After this, update the package, redeploy a new image to it, and everything should be up and running!