An easy way to cut down on your sugar consumption is to take out the sugar, and add in a carrot! While carrots themselves do contain a couple grams of sugar per serving, they also come packed with nutrients, including Potassium, dietary fiber, and Vitamin A.
An easy way to incorporate carrots into your baking is to shred them up. Using shredded carrots makes them much easier to use in baked goods, and will give the finished product an added texture and unique taste.
A second easy way to get the good stuff from carrots is to juice them. Get out that Magic Bullet you bought from that infomercial whilst drunk at 3 a.m., or whatever food processor you have available, and get a-juicing! Use the juice and pulp as a replacement for any water or oil your recipes may require.
The article linked has more information, including what is above. The final trick with subbing carrots for sugar is the hardest to deal with: your whiny family. Personal solution from me: don't tell them. If they ask, just lie. Convince them they're crazy.
"These cookies taste weird, Sam". - Little Sister
"No they don't. Maybe it's just your Dyslexia skewing your view of the world?" - Me
See? Just be blunt and change the subject. After all, you're doing them a favor by leading your family to a healthier life!