You might want to check out the Columbia Gem Motel. Most of the "motel" rooms are actually cozy cabins, the price is very reasonable, and it's a bone's throw from Columbia State Historic Park, which is dog friendly as long as your dog is on a leash and stays out of all the old-style businesses. (No sipping sasparillas inside the saloon, in other words.) The website for the Columbia Gem isn't very informative but here's the link: http://www.columbiagem.com/business/gem/
My book lists lots of great Gold Country getaways, should this one not work out. Write back if you need more suggestions.