Have you tried using Volumes?  

https://docs.databricks.com/en/connect/unity-catalog/volumes.html 
You can do it through the UI, on the Catalog Explorer > Add Data button. 

Also, you could double check if your workspace admin has disabled DBFS access, because you should still be seeing it on your catalog explorer (a button called "Browse DBFS"). It should be enabled, like this. (Settings > Advanced > Other)