- Pioneer Park is less than 3 acres, and one side of it is right next to people's backyards; I don't know if it would be the best choice.
- Stella Olson Park might be too wetland-ish, as a stream meanders through it.
- Sunset Park has enough free space (about 6 acres), but not many trees.
- Woodhaven Park looks similar in size and character to Sunset.
So Sherwood's City Parks are not ideal; Sunset is probably the best bet, even if it wouldn't be the most exciting DG course. However, they've got a whole lot of undeveloped space that isn't official designated as "park" but might be public land -- for example, along Cedar Creek, or in the wedge between Edy Road and 99W, or between the railroad tracks and Oregon Street on the east side of town. You might see if there are any options there.
You might also talk to St. Francis Catholic Church and the church just to the south of them; St. Francis has at least 4 acres of heavy woods on the east side of their property.
My sources for all of this:
Another option, of course, is to look outside the Sherwood City Limits, but you said you'd already had some interest from the City, so run with that for now...