Floppy carrots don't need to go into the compost. Use water to bring them back to life and keep them crunchy.
Bring limp carrots back to life by placing them in ice water until crisp. Better yet, store carrots in water in the fridge to keep them crisp. Change the water regularly and trim off the tops first, if necessary.
Do you just have too many carrots? Consider making this healthy and hearty Carrot Bread that everyone will love.