I see I'm the first to answer your post and I sure wish I could give you the information you want but I can't. I have Crohn's disease, with a very restricted food list, along with Acid Reflux Disease so cutting out carbs for me would just about take away everything that fills me up. Instead of asking for low carb foods you can eat, try posting a new thread and ask for the names of books you can read or websites you can visit to get the information you need. That way you can create your own grocery list with foods you like to eat. You may even find the information you want from one of the links on this website.
As for veges and fruits that rot before you can eat it all, ever think of trying frozen? They don't rot and you can steam the veges to keep them a bit crisp and just thaw the fruit when you want it. Just buy a few fresh items that you know you'll eat within a week. Unfortunately fruits and vegetables don't have a very long refrigerator life. Maybe 1-2 weeks at the most. My husband eats Red Delicious apples and they seem to last 2 weeks in the refrigerator and still stay crisp.
Wish I could be more help but with Crohn's the Carb diet is definitely not possible for me. Good luck on your search!