@Sophiegiraffe, I had three floors before my last move, five DCs, and my own private washer and dryer in the basement.
Here's how I managed:
Small basket in each bedroom (four small baskets on top floor and one basket in first floor/ ground floor bedroom). Five baskets total.
One basket at the foot of the basement stairs - stairs went straight down, no corners.
I took all the small baskets when they were looking full and sorted them quickly. I often only sorted into 'white' and 'other'. This saved a lot of time and meant laundry turnaround was faster. 'Other' went in a 30 degree wash.
I often bunged whites (mainly school shirts, socks, underpants, washcloths - small items) into one of the small baskets and threw the entire thing down into the basket at the bottom of the stairs, with a few bathroom towels, kitchen towels, my kitchen aprons, baby bibs etc and washed them even without filling the washing machine. All the towels in my home are white.
Meanwhile, all the non white items could also go into the washer together - jeans, t-shirts, sweaters, hoodies, uniform trousers, pinafores, skirts, baby clothes, etc. Again, I just threw them downstairs to the basement, sometimes baskets and all. 30 degrees, job done.
The downstairs basket was brought up full of dry clothes with any smaller baskets stacked and carried on top. Clothes were folded and returned to bedrooms inside small baskets.
Then the basement basket went down again immediately.
I hardly ever carried a basket down to the basement, and certainly never carried one back up again with dirty clothes in it awaiting their turn.
Try doing darks and colours together at low heat?
If you need to separate darks and colours, buy enough baskets to hold each and will hold a full load so you won't end up carrying dirty clothes back upstairs?
Wash more frequently?