>Do you only have that one sink basin with the disposal attached?

No, there are 2 sink basins, each with its own p-trap: https://i.imgur.com/6LGoz9q.jpg. Looks like the second one is correctly installed

FWIW I've seen signs that the previous owner of this house did a lot of DIY. I'm thinking he may have installed the disposal himself without fully knowing what he was doing

Gotcha, yeah I guess it technically does function as a p-trap. My understanding is that ideally, only the bendy pipe should hold water instead of the vertical pipes.

I was also thinking about what caused the old one to fail in the first place, since the washer on the lower part had completely detached from the pipe. My thought was maybe it's holding more water weight than it was designed for? Could the friction fit be failing because the column of water is too high?

Thanks! Thought so. Yep you're right about the vertical drain pipe through the floor, I'll look into doing that