Is there a dog park around there or a place the dogs can run?

There is a dog park near there also. Basin Homes Dog Park http://www.cityofml.com/facilities/facility/details/basinhomesdogpark-53 Fenced, off-leash dog park with separate areas for large and small dogs Location 1201 N. Paxson Drive, Moses Lake, WA 98837 Phone Number 764-3805 Features Benches, Tables, Dog Play Equipment

It's a park with some open space. If the dog is well behaved and you clean up after them, they have room to run around.