As @Ryan says, if you re-frame your question / requirement then you can achieve what you need by creating and editing your documents in a collaboration site document library with major and minor versioning enabled, and visitors limited to view major versions only. Then you can edit without anyone else seeing your work until you decide to publish a major version.
If you truly want to move documents between libraries, sites, etc, then you have basically two options: Explorer View, or a 3rd party solution.
The ability for end users to move their documents between sites, libraries or folders within a single library is a significant gap in the out-of-the-box capabilities of SharePoint, neither 2013 nor 2010 have this in the web GUI. The only way to achieve your requirement with OOTB is using Explorer View.
The other suggestions don't cut it in the OP scenario, where the end user wants an experience similar to Windows File Explorer. Users want something that is easy to do and seamless with their day to day activities.
If you are not familiar with Explorer View, click on the Library tab of the ribbon bar and look in the Connect & Export group. It might be disabled if your farm admin has not enabled or has turned off certain capabilities. Also, Explorer View has its own quirks, as illustrated in the blog post SharePoint and the “Cannot move : Cannot read from the source file or disk.” error via Windows Explorer View.
If you want a really easy, clean, seamless experience for your users that is part of the web GUI of SharePoint collaboration sites and document libraries, check out the 3rd party ISV market. As an example, SharePoint Products CopyMove for SharePoint 2013 does exactly what you need.