Is this enough to loose 20 pounds? How long will it take?

Okay, so I am a 13 (and a half, ha-ha) and slightly overweight for my height. I weigh 150 and I want to be around 130 or less. I've been avoiding sweets and grains, I never drink sodas, I drink mostly water, and I am reaching for more fruits and yogurt and stuff. Everyday I run around our 1 and a half acre yard, do 10 push-ups, 15 crunches, 30 lunges, and some stretching. I don't have time for much else due to the fact of school, writing a book, pets and my family and social life. I found a few videos I was going to try. But will this even help? Also, back to eating, I am not a picky eater, but I am very sensitive, (My Mom said it's some sort of sensory disorder?) and I cannot hear certain noises or eat certain textures, so a lot of fruits and veggies are out of the picture. I can, however, eat bananas, strawberries and peaches, but I cannot eat things such as apples. Anything suggestions, foodwise?