I don't have a newborn, but I do have a 2 and 3 year old...and it's shocking how much time the laundry takes up. And I agree -- the toughest part is putting the clean clothes away. If I let too many loads of clean clothes pile-up, it can get tough to deal with. Everyone's different, so here's how I have learned to manage without losing my sanity:
1) The kids have two community hampers in the hallway -- one for jeans/darks and one for light colors. At 3, she is old enough to learn which hamper to use...and the 2 yr old is learning from the eldest! Putting their clothes in their hampers is part of their bedtime routine and is non-negotiable. They think it's fun and love the positive attention when they get it right!
2) My husband and I use a three-section hamper (lights, darks, and his business shirts). (I keep my delicates separate and clean them with the kid's lights.) It takes up a ton of room in our closet, but it has been seriously worth it.
3) I do a load of laundry every day or two. Adult lights can usually be combined with kid lights, etc. However, I do NOT sort and put away until I have 2 or 3 loads clean.
4) Once I've got 2 or 3 loads (but not much more!) I let the kids do the sorting with me, usually in the afternoon after the 2yr old gets up from nap. They LOVE it! One basket for Mommy/Daddy clothing, one basket for the 3 yr old clothes, and one basket for 2 yr old clothes! Once that's done, I have them sort Daddy's socks by colors/types! Just call it the "matching game!" Again, your kids will love getting positive feedback and attention from a successful "job!"
5) Once the sorting is done, everything gets put away right then and there. My own non-negotiable is that I have to get all the kid's clothes put away at minimum. While I'm putting the clothes away, the kids play upstairs with the upstairs toys. Upstairs toys are not allowed to come downstairs...so they feel like there is variety! And we are all together in the same rooms.
6) And on those occasions when things get backed-up and out of hand (yep, it will still happens), it's time to call in the hubby for back-up. He can handle putting his own stuff away once in a while.
There aren't many projects that we Mom's can "check off our lists." When ALL the laundry is clean and put away, there will be dirty clothes at the end of the day. When all the dishes are clean and put away, there will be a dirty dish or bottle within hours. It can be frustrating. But hang in there...the rest of us are feeling it with you!