I watched a few YouTube videos on potential issues with my doorknob. They suggested that it was likely just sticking, and some WD40 could fix it up. So, I took it apart... and I don't think that's actually true. It seems that the metal parts against which the door knob pushes in order to push the latch have worn down -- it's hard to know exactly how much, but at least 2mm on one side, if not more, which is enough to cause the latch to stick.
This means that there's no way for the knob to turn the latch far enough to not catch; the backstop is just too worn down. (The door knob appears to be many years old, and looking at the quality of the construction, and the lack of the brand name anywhere on it, I'm thinking that it was a pretty cheap knob in the first place.)
So, I went out and bought a cheap Schlage lever knob to act as a stand in; it's not a real replacement, because it's not an exterior doorknob (no lock), but at least I can get in and out of the door again.
I also wrote an email to the landlady which I should have written a week ago, asking to change a specific term of the lease, and letting her know she'll need to buy a better doorknob for me to put in whenever she gets a chance.
So, at least I did get something to go right.
Of course, the stress over the door and the email I just sent woke me up at 4am. So, you know, you win some, you lose some.