Having lived for many years in Salem, we "dry" camped many times in the areas you are planning on visiting. Along with what others have mentioned, some more suggestions:
First, look at the Nat Forest sites for the area, many semi-developed campgrounds available, and a lot are free.
Look into the "Century Drive" area out of Bend. Lakes, rivers, and dispersed camping.
Around Sisters, head toward or up to 3 Creek Lake. If you can get to the lake (depends on snow levels, but looks like they will open early this year!!), there 2 NF campgrounds right at the lake, and one just before at 3 Creek Meadow. Also, lots of unimproved, dry campsites almost anywhere off the main road up to the lake and beyond. A decent one is "Black Pine Springs".
Near Camp Sherman are many dry sites in the Jack Creek Area. You could also drive a bit more to the NE and visit Lake Billy Chinook and Cove Palisades State Park. Busy, but nice.
On the way to Eugene, there is Clear Lake Campground, and quite a few others on the road over the pass.
So many more, but my memory is slowing down!!
Enjoy your summer trip, you will love it.
