Is there any way I can avoid Boirer and put my laundry on Shabbos in its specified location of the divided hamper (dark, color, whites), for example by spreading out the laundry and putting one item at a time? Or do I have to wait for after Shabbos to put it in the proper division?
Organizing the laundry was on Shabbos is not only an issue of borer, but also an issue of ממצוא חפצך, doing something on Shabbos to prepare for a Melacha taking place after Shabbos.
The issue of borer can be circumvented if you throw the pile of clothing in such a way that there will no longer be a mixture of clothing, rather distinct garments spread out across the room.
The issue of ממצוא חפצך can only be avoided if such behavior does not appear to onlookers that it is in preparation for a Melacha after Shabbos, or if the post Shabbos activity could be done in a manner which is not a Melacha, or if done for a Mitzvah.
It is difficult to see how this would be the case with the separation of clothing into different hampers.
See about borer here.
ממצוא חפצך – שו”ע סימן ש”ו סעיף א. שם ג. אדה״ז שם א-ג. שם ה-ו.