Looks N Fork Smith River and Smith River tributaries, Calapooya, and Cow Creek in the Umpqua drainage are where you can fish for trout and keep 2 per day. I didn't search much, but Douglas Country streams seem to mostly be in crisis.
And bait fishing for trout in streams is pretty much not possible these days.
If you travel north on I-5 one county, you'll find more friendly streams for trout fishing, even if you want to use bait. Heck, very close to Douglas County are the headwaters of the Coast Fork Willamette... I found fishing from London Springs up to where it splits into Big and Little to be quite pleasant and full of native cutthroat. Above the falls on Big, the fish are small, but still plenty of fun and you just might find a large one up there.