What I am doing wrong ...
In principle you are not doing anything wrong, you simply have a wrong expectation what the menu function "Save as.." actually entails/does.
Save as /only/ saves the project file (and it's settings) but not the units that are part of your project. If you wish to do it that way then you need to (re)save every individual unit that belongs to your project to the new location.
That is also why the compiler is unable to locate the unit, as that still resides in the old location on disk (and the compiler does not search for files in the old location, unless you specifically tell it to).
afaik you have two options:
1) use version control to safely store your current work in a repository (requires you to install/use/familiarize yourself with svn/git/mercurial/cvs/etc)
2) as a easy alternative you can use publish project, which will take all your files belonging to your project, compress them into a zip file and store the resulting zip file to a (preconfigured) publish directory.
Both methods allow for safekeeping all your files belonging to a project and in which case you do not change the location of your project. by using the menu-item save as.
If you wish to keep things /really simple/ then you can also copy your whole project directory to another location (you can use a proper file-manager for that)