Why do carrots spoil so fast?
First we will take a look at the lines of resistance of a carrot. There are three


  The skin - The first line of resistance is the outer membrane or skin of the carrot and chemical compounds that are normally present
  there. Two antifungal polyacetylenes, falcarinol and falcarindiol, and
  an isocoumarin, 6-methoxymellein, are present in small amounts in a
  normal carrot and present the first barrier in undamaged carrots.
  
  Post Harvest - If during harvest or post harvest handling this barrier is broken, the surface is punctured, scraped or sliced,
  microorganisms may begin to grow. The carrot at this point begins a
  wound healing process by accumulating suberin, a complex
  polysaccharide, at the wound site. Along with suberin, lignin is also
  accumulated. Suberisation is complete within 48 hours after wounding,
  and lignification is complete within 168 hours of wounding..
  
  Phytoalexin barrier - The last and most important line of resistance is the phytoalexin barrier. Phytoalexins are chemicals
  produced by plants that impart disease resistance in response to
  mechanical injury, physiological stimuli or infection. Scopoletin, a
  coumarin,and para-hydroxy benzoic acid, a phenol, are apparently early
  arrivals on the scene but later disappear (reference).


There are three enemies of carrot storage: wilt, re-growth and rot. 

Some reasons for carrots rotting faster: 

Many fruits and vegetables start to produce ethylene gas when they start to ripen. Some foods are extremely sensitive to ethylene gas which makes them rot faster (reference). I am guessing that you keep different vegetables close together. Maybe if you stored carrots separately, it might make them last a bit longer.


  Major post-harvest losses in carrots are due to Sclerotinia rot,
  Botrytis rot, Bacterial soft rot (Erwinia) and Sour rot (Geotrichum).
  Of these, Sclerotinia rot is the most prevalent.The Sclerotinia fungus
  infects the carrot roots in the field through the crown. During
  storage, the Sclerotinia fungus produces a white, cottony mycelium
  which covers the roots. Infected roots are usually soft and watery (reference).


There would definitely be more reasons but I am guessing that these are the major contributors to it rotting faster.