Using this was a long shot, since it's old and has few users, but I hoped it might do what I'm looking for. But no. I does not delete data stored in DOM storage or IndexedDB. Maybe the developer will see this and update it. One can only hope ....

Incompatible with ScriptSafe.

The add-on description says "The addon does not touch the IndexedDB databases created by other addons, only those which were created by websites", but it actually wipes out all ScriptSafe settings :/

Edit: I don't know how ScriptSafe save their settings, but I know that whenever I have both ScriptSafe and StorageErazor enabled, my ScriptSafe settings get nuked every time I restart Firefox. Turning off StorageErazor solves the issue. None of my other extensions seem affected by StorageErazor.