Moldy Home Made Bread

I have a wonderful bread maker that makes a 2-lb. loaf of bread. I use bread bags to store it. It always gets moldy by day 3. I use honey in my recipe instead of sugar, which I'm told is a preservative, but it doesn't work. I've tried ascorbic acid and a couple other things I don't remember now. I've also tried storing in a paper bag, which just dried it out. The best thing I've found is to slice it and put it in the fridge. It's to cold in the freezer. I take out whatever amount I need and put the rest back. But I don't like it cold. I'd much rather be able to use it at room temp.! Any ideas, suggestions? I'm beginning to think it may just be one of the pitfalls of making homemade bread.