carrots washed and dried rot within the week

I suspect the washing allows more water to penetrate and a different range of bacteria. the soil doesnt have these bacteria present.

When carrots are commercially washed it is a tumbling process with very little water - the carrots then go into a cold store to keep them for months on end if necessary. If you or I scrub our carrots and then leave them to sit around at room temperature they will rot. Immature carrots will rot after being pulled in any case if not used within 24 hours.
Carrots are one of the crops that can be left in the ground until needed unless there is a hard frost. Even then if under fleece they will be fine.