Nearly all Oregon State Park campgrounds welcome dogs. Pets must be confined by the owner, or on a leash not more than six (6) feet long, and kept under physical control at all times. The closest State Park campground to Florence would be Honeyman State Park.
The following link is to Oregon State Parks brochure on pets in parks: http://www.oregonstateparks.org/index.cfm?do=main.loadFile&load=_siteFiles/publications/pets-in-parks.pdf
The following link is to Oregon State Parks “Find a Park” page and you can check all of the State Park campgrounds south of Florence: http://oregonstateparks.org/index.cfm?do=visit.dsp_find
There are also countless private and Forest Service campgrounds that are dog friendly, but rules can vary.