I've created some Azure Machine Learning Workspaces and associated them with "classic" storage accounts; but would like to have them associated with "not-classic" (or whatever the term is) storage accounts.
Is there a way to convert the storage accounts from "classic", or to change the storage account associated with a Machine Learning Workspace?
You can not convert classic to resource manager. The best way to move to rm is make a rm storage and copy the content from the old account.
As of today, there's no automatic way of converting a "Classic" storage account into "Azure Resource Manager (ARM)" storage account. Today, you would need to copy data from a classic storage account to a new storage account.
Having said that, there's no difference in how the data is stored in both kinds of storage accounts. Both of them support connecting via account name/key and/or shared access signature. The difference is how these storage account themselves are managed. In ARM storage accounts, you can assign granular role-based access control (RBAC) to control what a user can do as far as managing the storage accounts (like updating, deleting, viewing/regenerating keys).
Regarding your question about using new storage accounts with ML workspace, I don't think it's possible today (I may be wrong though). Reason being, ML is still managed via old portal which doesn't have the capability to manage ARM storage accounts.
After the question was asked and answered, Microsoft provided a way to achieve this through PowerShell. I have successfully used the steps in this guide to convert Classic resources to the new ARM architecture, including Storage accounts (detailed near the bottom of the article).