Permissions issue with Godaddy virtual dedicated server

I’ve recently upgraded from shared hosting to a Linux virtual dedicated server with Godaddy. In order to edit the theme files in WordPress or upload media I must now adjust public permissions to writable (from 644 to 646).

I’ve read a bit about the issue in other forums, but the only solution suggested is to adjust file permissions each time I want to edit the theme. What can I do to solve this problem leaving the file and folder permissions at their default settings (755 for folders, 644 for files)?

The site this currently affects is patriciacore.com. I plan to move more sites over to the new server once I figure out this issue.

Any assistance is appreciated.