Two steps forward, two steps back
After a scarry 45 minutes of having no connectivity at all on two of my computers (including being able to even access the router. Luckily the old NT box never lost connectivity at all and could get to the router) I was able to restore Internet connectivity to all of the computers. I did accidently make some progress however (I say accidently because, honestly, I have no idea how I did it,) because the two desktops can talk to each other (though that feature’s always been there) and the laptop can see the two desktops. Unfortunately, neither desktop can see the laptop, nor can the laptop actually access anything on the desktops. I think I’ll stop worrying about this problem for the night and just continue installing software and updates on the laptop.