DC resistance will be around 200r, so it probably will not show continuity, but resistance. Most common fault is that the coil is fine but the magnetic impeller inside is worn and sticks in the housing. The coil, magnet and impeller are available as a spare part on their own, and just are a few screws ( 6IIRC) to remove and replace on the housing. There is a o ring seal that you need that will come with it, lubricate the seal with some soapy water to get it to fit, and put together.
The kit cost me $15 last time I got one earlier this year, it fits a lot of washing machines.