We've just spent the weekend camping at Roundhill campsite in the New Forest. It's where a WWII airfield was so it's pretty flat and a lot of the old concrete roads still exist so walking can be as easy or arduous as you like. The airfield was enormous in its day. The campsite welcomes dogs (maximum 3), they're expected to be on lead in the (enormous) campsite but you're rarely more than a short walk from the forest proper where they must be under control but can run free. There are poo bins and a dog shower as well as all the usual people things. Beaulieu is a short drive away with Hatchetts Pond for doggie swimming and Stephs Kitchen in the very nice Garden Centre. Dogs are welcome in the garden centre and at the outside tables in Stephs Kitchen. You have to order inside though so it can be tricky if you're on your own with dogs.