Field crops (carrots and more) are cooled as soon as they are picked/harvested during the cool hours of the early morning..

Cutting off the tops will also help.

Not an expert but heres my expereience:

Some carrot varieties keep better than others. I grew nutri red carrots this year and last and find that they dont keep well. They go soft quickly (im talking just a couple of hours) and look a bit like yours in the photo. The orange ones kept much better. And yes, if you cut off the tops and store them in a cool place, they will keep much better.