They said something about me having to take out the printing head of
the old one when the replacement arrives (I wasn't clear on exactly
what that meant, or maybe I misheard). That might mean they're shipping
a 4000 unit that doesn't have a head. But they'll be shipping it
Thurs., so I'll be able to relate the conclusion soon.

You should phone them back asap, and explain you can't remove your old
head.
This leads to a new printer with a new head and new ink, and unless you
take a hacksaw to your printer, you in all honesty can't remove the old
head.

The alternative is phoning them when you get the referbished version
and telling them you can't remove the old head, and waiting a couple of
days for a new one, but odds are you wouldn't get the free ink. This
is assuming they will honor the warranty, which it looks like they may.