Free: Contests & Raffles.
So the first and most important part of your plan is that shed hunting in the park is prohibited. If you get checked by a park ranger with sheds you're in trouble.
Hard to recommend due to not sure about your skills, abilities, and tolerance for company(or lack of company). I would recommend looking at WTA website and pick something that sounds interesting/doable. Mountain miles - even on trail - come hard when you’re new with a backpacking load. I’d suggest keeping it short without too much elevation gain. More of a drill on win self-sufficient and way to hike out if some gear etc. doesn’t work out. There are some great places to visit less than a mile or 2 from the truck. Many are less crowded than the high alpine meadows many miles further in.As for elk, you’re way more likely to see them on the E. Oly peninsula driving to the trailhead then in the backcountry in my experience.