I feel your pain @Bshag4lv, my last generation of my current game had 4 children plus the 2 adults, and laundry had to be done every single day.
How I handled it was, I had 2 hampers in the house, each in one of the bathrooms. In my game the laundry room is just before the kitchen, so I just had someone each morning bring clothes to the washer after taking a shower when they were on the way to breakfast. With 6 sims in the home, every shower got used, so it was never even having them go out of their way. Just a simple matter of shower, grab laundry, drop it in the washing machine, then go eat breakfast. Whoever brought the 2nd hamper is who I had start the washer (you can put multiple hamper loads into a single load of wash). Then the last one out of the house for work would move it to the dryer and start that.
Basically it was laundry day every day, but doing it that way, it was never more than a minute or two extra time seeing as every trip was one where they would have been going past the washer/dryer anyway. And doing it every day made it so neither hamper ever got "full".
Another option if the above scenario doesn't work for you for some reason, is to install mods that make laundry work how you indicated you want it to...which is gathering all of the hampers before going to the washing machine. LittleMsSam's small laundry fixes does that (and more): https://littlemssam.tumblr.com/post/175411944173/small-laundry-overhaul-a-small-overhaul-for-the I don't use the mods at the moment, but have used it just to see how it works and it does what it says very well. The one you'd want is the one that lets them put the laundry into inventory.