I think I have a different system compared to most houses I hear about and see.
When I was growing up, there were no hampers or baskets in rooms- everyone was expected to take dirty laundry right to the washer/dryer area. I honestly didn't even know about hampers until I got older. So I continue to do the same thing in my apartment now. I can't imagine hampers taking up extra space just so people can store dirty laundry in their rooms. So anyway, we take laundry straight to where the washer and dryer are (which is in a closet in a hallway). There is room on top of the dryer for 1 basket for clothes/linens and a small bucket I throw diapers in. If the laundry pile gets any bigger than that, it has to go on the floor in the hallway, and we'd have to step over it, so it's good motivation to go a load every time the basket is getting full.
Also, less clothes, towels, etc mean less can pile up. So that is making a big difference for us. If you have a lot of baskets or hampers, then that's just more room for dirty laundry storage. I imagine feeling pretty overwhelmed thinking about multiple baskets waiting to be washed.
Maybe try catching up, then switch to less space for it to accumulate. Does that make any sense? lol maybe I'm just weird!