PowerShell does not automatically save anything to $pshome. Most commands that save something, such as Out-File, simply default to the current directory, whatever that might be. You can always specify an alternate path, along with the filename, if you want something saved elsewhere.
I presume you're using the ISE, because you mentioned "save to." The ISE also defaults to saving to the current directory, whatever that is. It doesn't default to $pshome. Setting a variable doesn't change any of that behavior. If you want to save a script to C:\Script, just type "C:\Script\" before the filename. Same as anywhere else in Windows.
The ISE may default to your Documents folder – like most other apps do – but you can specify any other directory.