I know opinions differ on this, but I usually think
of a swale as a tree-planting system, and it is dead level on contour. So what you're planning I would just call a ditch or a channel or a stream.
To avoid erosion completely (always a good idea), I would generally keep the fall of your channel to 2% max, and preferably closer to 1%. If it's heavily planted with water-loving plants with good root
systems, you could maybe get away with a bit more, but only once those plants are established, so you'd have to devise a way to keep the flow gentle in the initial stages. Do you know what the maximum possible flow is out of that spring? Combined with a 100-year rain event for the area? (Like hurricane leftovers blowing through after 3 weeks of solid rain, or some such thing.) That's what you want to plan for.
Meandering around the property gently like that it can do a lot of good. I'm sure there's a limit to how much digging you want to do though, so if you want to drop it down a few feet at a time, you might want to create some infrastructure. Maybe vetiver grass or something if you want to go all-natural.