Thanks to the ever wonderful Erick Laabs at The Mac Store, I tried plugging my iPod in and told iTunes to not sync the music library.
I was then able to select the iPod and choose settings. Then on the music tab I selected sync manually.
Now I can drag things to the iPod or delete them off!
Podcasts added in this manner do not show up under Music->Podcasts. That is reserved for the synced special playlist. But they do show under Music->Genre-Podcast.
I also added a dumb playlist called “Added from Laptop,” drug my laptop music into that playlist, then that playlist to the iPod.
Once added, music is there for good until you either manually delete it or do a full sync to a library.