For Firefox I recommend Cookie AutoDelete 3.8.2 addon. It is possible to configure it to erase all storage (cookies, indexeddb, localSorage and others) after chosen time (default 15s) after changing/leaving domain. It has many other useful options.
Another good addon is ClearURLs 1.27.3 which clears suspicious parameters of query strings on everything what's loaded by page. It sometimes break sites.