Hi, if I'm following correctly, you should be able to point your existing ArcMap install to the AGOL credentials for Org B.
First (with ArcMap closed), open ArcGIS Administrator. click Advanced. Click Manage Portal Connections. If necessary, add a portal connection to https://www.arcgis.com. Click Connect. Like this:
Next, start ArcMap. Click the File menu and click Sign In. Enter your Org B credentials in the sign-in window. At the bottom, check the Sign me in automatically box so you only have to do this once.
I think that should work.