Unless you need $$ income, I'd recommend focusing on State Parks. Usually the workamping opportunities there are in green, spacious areas with little to zero dog restrictions and lots of space to hike w/ the paws. Typically no pay, but you'll get a campsite in return.
Almost every State Park in the country has jobs. So, look at individual opportunities in areas that interest you.
We workamp ~2-3 months a year as lighthouse hosts on the OR Coast (for OR State Parks). We get a full hookup site in a beautiful beach park (that's completely dog-friendly) and only work ~3 hours/day. Totally do-able with pets!