What about the Grand western canal in tiverton? it runs from tiverton nearly to the somerset border, loads of walking and lots of places to stop en route either in a local pub or with a picnic.
Another suggestion would be Killerton near Exeter, lots of grounds to walk around, and the house too....plenty of room for a picnic.
Also haldon forest (top of haldon hill near exeter) loads of forest trails and great views
I know its almost in Dartmoor - but not quite - a favourite of ours is Fingle Bridge not far from exeter and near castle drogo...loads of riverside/forest walks with a designated picnic area next to the river with toilets and if you did fancy a drink you have the lovely riverside pub.
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