It's actually really, really easy to get the state of a "dumb" washing machine (or any other electric machine) into Home Assistant using a smart plug. You can use something really basic like "power draw for > n seconds followed by no power for > m seconds" to detect when a cycle is finished. You can get way fancier and look at power draw curves to determine what part of a cycle it's in, or which cycle, if you really want to. You can add a door sensor (recommend Samsung) if you want to know if the door's been opened.
Unfortunately it's much harder to do the same for an electric dryer, since there's no inexpensive or good smart plugs for 240V last I checked.
You can use a vibration sensor on the dryer.
Unfortunately, as you noted, I haven't figured out how to handle the dryer as the load is too much for all the smart plugs I've found. I wish there was a clamp-style monitor on an extension cord, but it seems that is something which doesn't exist.
Thankfully, the dryer isn't as much of an issue since dry clothes can sit there until I remember to get them.