On my local course, which is in a private state park, the only way they would let me put Trash cans on the course is if I emptied them myself. Which I have to do about once a month for 3 cans, the course is probably the cleanest I have ever been too. Of course it is in a state park, with a gate, which doesn't get near the traffic that a lot of courses do. My point is, pick up what you can, and feel free to take the initiative and carry out a couple of trash bags when you can. Sure no one will notice, you won't get any atta boys, but if you are sensitive to a beautiful place (a disc golf course in my eyes) being tarnished by the site of trash, then do your part for the game.
What is a "private State Park"? Not busting balls, I'm serious.