This person already worked out quite a bit of the mechanics of integrating all this with Home Assistant, so I will probably copy a lot of their work. I haven't yet analyze all the machine state stuff yet, so I don't know if I agree or disagree with what they decided.

(some time passes)

I'm definitely going to take advantage of that person's mechanics in  how they set this up. My notion about the machine states is a little different. It's more detailed, but it will probably turn out that I don't need most of the states, but I'm going to start with all of them and see how it goes. I also expect to organize my automations differently, but that is mostly a style matter rather than a fundamental difference.

First up, the dropdown helper. I had to consult Home Assistant documentation to find it in the UI because it has moved since that write-up. It's now at Settings > Devices & Services > Helpers. I originally planned to use the same states for both the washer and dryer, but it seems the helper is really an individual entity rather than something that can be associated with some other kind of object.

Here's the helper for the washer. I have an essentially identical one for the dryer. There is probably a smarter way to do this that does allow sharing, but the duplication is OK for this simple project.

In addition to the states mentioned in a previous project log, I also defined an "Unknown" state. That will be used when Home Assistant starts up since any number of events may have been missed. It will also incidentally be used the first time I start all these things going.

The UI doesn't have the usual option to view or edit the YAML. I found it afterwards in file config/.storage/input_select, which is referenced from config/.storage/core.entity_registry.

{
  "version": 1,
  "minor_version": 2,
  "key": "input_select",
  "data": {
    "items": [
      {
        "id": "washer_state",
        "name": "Washer state",
        "icon": "mdi:washing-machine",
        "options": [
          "Unloaded",
          "Loading",
          "Loaded",
          "Running",
          "Stopped",
          "Unloading",
          "Unknown"
        ]
      },
      {
        "id": "dryer_state",
        "name": "Dryer state",
        "icon": "mdi:tumble-dryer",
        "options": [
          "Unloaded",
          "Loading",
          "Loaded",
          "Running",
          "Stopped",
          "Unloading",
          "Unknown"
        ]
      }
    ]
  }
}