Leave the WordPress files where they are. You can "Move" the site by just moving and editing a file or two and changing a setting in the Dashboard - but please READ and understand the instructions before you do anything or you will be back here in a PANIC.
The link that esmi gave you has good instructions - they look worse than they really are.
If you have questions ask them here BEFORE you do anything.
Old links work just fine.
Remember to leave an unchanged copy of the original index.php file where it is so someone coming into the /blog folder will not have an error -
I "Moved" my site into the root directory about a month ago - I followed the instructions - scared the daylights out of me but the final shuffle took less than 5 min. and worked great. Now my whole site except for a photo page runs off of WordPress. Much Much better than struggling with HTML when I want to make a small change. I now add new content regularly for boating safety.