Hi and Welcome to the boards breambasher, I have only been to Oldmoor twice and its a nice place sadly I don't know were you can walk the dog but this reply will bump it to the top again :)
Hi Breambasher welcome to the community you can walk your dog at Adwick washlands as it is an open site with a bridle path/ footpath running through the middle of it.
There must be public footpaths at Bolton ings www.rspb.org.uk/.../about.aspx
Gypsy marsh www.rspb.org.uk/.../about.aspx
Wombwell ings www.rspb.org.uk/.../about.aspx
thanks alan will try them out
I've only been to Adwick other than old moor so don't know what they are like but at Adwick you can do a circular walk.