Ways to Store Carrots for more days
I have worked in the CPG industry for almost 30 years. A few years back I had the privilege of representing one of the largest produce brands in the US. I learned quite a bit about produce storage and what speeds up the deterioration rate.

One thing I noticed in your picture is that there appeared to be moisture (condensation) inside of the bag of carrots. It is certainly okay to leave them in the bag they were purchased in, but you want to be absolutely sure that there is no moisture in the bag. Moisture can be a real devil when it comes to mold and mildew or other fungus.

Another very important thing that affects the length of time produce will remain good is air circulation. You have to have really good air circulation around produce. Bagged salads, slaws, etc. will go bad quicker (even if they are unopened) if they are stored so that they don't have proper air circulation around them. In the case of the carrots, I would recommend opening one end of the bag so that they can get air.

Overall, I would rceommend that produce be stored in the produce crisper in your refrigerator. Crispers are designed to maintain an appropriate temperature and level of humidity that will help produce stay fresh longer.

If your refrigerator does not have a crisper, I would get a plastic rectangular container and place it on the bottom shelf of your refrigerator to use as a make shift crisper. If possible, cut one inch holes generously in the lid to allow for air flow. If you don't have a way to do this, leave the lid off.

Regarding using the carrots, I would be reluctant without knowing with certainty what the black is and probably would not use them.