So I am just testing out what the difference is between using Sharegate and manually migrating SharePoint 2010 to 2013. Reading a walkthrough, it says to move all SharePoint databases from the sql server to the new sql server and re-attach the databases. I know of the WSS_Content and Sync DB, Social DB and Profile DB, are there any other databases I need to move? Also it says that I need to move the log files as well, but when I try I receive an error saying that the file is already in use and so I cannot copy any log files.

I need

The article below is about SharePoint: Manually upgrade Service Application database.

http://blog.jussipalo.com/2014/01/sharepoint-manually-upgrade-service.html

Also, have a look on this article which includes a list of common issues and describes general principles to help you identify and address issues during an upgrade from SharePoint 2010 Products to SharePoint 2013
Upgrading from SharePoint 2010 to SharePoint 2013 Step by Step
As a workaround create a site collection in SharePoint 2013, detach the newly created content database and then attach the 2010 content database to the site collection. SharePoint 2013 will upgrade it for you. The process is explained here

Moreover, If you wish you may also try this automated solution which lets you to migrate from SharePoint 2010 to SharePoint 2013 without any interruption.

Hope it helps you!