Did what everyone should always do - called Moffsy.
She taught me how to drain all the water out of the tub (very easy by the way).
Then she told me to go Google my problem. I did. Turns out it was probably the lid switch mechanism.
At the same time I also Googled the price of new front-loading washing machines. Average price - about $1000.
She had me check the lid switch (it involved sticking a screwdriver in the switch hole. This doesn't sound safe but it was). Then the switch (which was already semi-broken) broke right off.
Yup, that was definitely the problem.
Went to the parts store that has every appliance part ever made and bought the new switch for $60.
Half an hour of quiet time this afternoon, a few screwdrivers and the machine is rinsing and spinning like new.
Thank you Moffsy, thank you appliance store, thank you Hops & Barley for watching t.v. while I did this, thank you baby girls for napping.