When I asked how to Differentiate Production/Sandbox Instance? I ended up following a strategy of parsing the username. Somewhat brittle, perhaps, but it should work for the majority of your users. If you add this field detection at the user level, then you will be able to merge
$User.Is_Production__c into just about any formula in the system.
You could certainly adapt this solution to your needs with a formula that renders everything after
@mycompany.com. as the approximate sandbox name.
RIGHT(Username, LEN(Username) - LEN("@mycompany.com") - FIND("@mycompany.com.", Username))
Then you can reference
$User.Sandbox_Name__c = "MyFullSandbox" in your formulas, for example. You could probably make this solution more robust by blending the approach with
Hierarchy Custom Settings, taking the most common value amongst all users and storing that as the org default.