Also at least in my experience the dryer takes longer to run than the washer, so if I've just rotated a previous load and started both at the same time I'm always waiting on the dryer to be able to do the next one.
My understanding is that detecting an electrical current through a power cord typically doesn't work when measuring all the wires together, because the current in the live wire is canceled out by the return current in the neutral wire. This is why clamp-style meters are usually put at the electrical panel, where individual wires are accessible. Since I live in an apartment, that isn't an option for me.
https://electronics.stackexchange.com/questions/634852/exter...
It's also possible that your machine lets off some elecromagnetic noise you can detect directly.
The other thing you could try is put some sort of a microphone / vibration on it, then train a small NN on the waveform to distinguish between the washer and 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.