I understand completely where you are coming from, since I was a Project Linus coordinator. It can be overwhelming. Simply hauling boxes and bags continually is an energy drain. Sorting it and having it taking up a lot of space is another real problem.
Do you have Linus meetings or work sessions regularly? If so, I would suggest bringing the possibly usable cotton fabric unwashed to the meetings. Tell people that it is as you got it, and if they can use or (or know of anybody who can), then they are welcome to take it home. Put it out of your mind once it is gone.
If you need to put pressure on the group, tell them that whatever isn't taken goes directly to Goodwill.
Another option is to recruit other members to store the fabric. For instance, one person takes a bin/box of blues; another one takes reds. Then rotate what color boxes are brought to meetings. Also, let the whole group know who has what color, so they can ask them directly for pieces.
If you decide to make kits, keep them simple. Otherwise you are in another time trap -- always cutting, never sewing.
Best wishes! You are in a challenging situation. The only was I could escape a similar situation was to move out of state - ha!
Dayle