Have you considered contacting ur local DSHS office to get a list of shelters? There are emergency shelters for parents w/kids. If you need help finding resources, repost on this page and I'll try to see about connecting with you.
There are no monthly rates. You are allowed to stay for a 14 day maximum. You can leave for a day or two and return for another 14 days. I would recommend DSHS as well. Good luck.
Not sure per mon. Price
Campsite accommodations Ph: (888) 226-7688 There are $25, and $12 per night sites. Not sure you will get a monthly fee as thwre are time limits for staying.
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