Carrots and green beans are great snacks. My Jaxx loves them. I usually give them to him after I have froze them because they seem to last a little longer that way and he seems to like them better.
Personally I would also look into another dog food. Science Diet is not a quality brand dog food and I think of it like giving a child McDonalds for every meal (basically junk food.)
There are some great top quality dog foods out there. Dog Food Reviews and Ratings | Dog Food Advisor
is a great site that gives ratings on dog food. I would personally look for a 4 star food at least and try to go for a 5 star.
I would recommend Acana, Orijen, or Fromms. If you buy your food at Petsmart (or a similar pet store) I would go with Blue Buffalo Wilderness, Wellness Core, or Innova Prime. There is also a great food called Ziwipeak that is freeze dried raw food that looks like jerky.
It sounds like she is not getting enough nutritious food and the fillers in Science Diet are just not enough to fill her up.
Dog Food Calculator
is also a great tool to give you a basis on how much you should be feeding her with the food you have. When you put in the weight don't put in her weight now but what should be her ideal weight.
Personally if I knew she was getting enough food and that there was nothing medically wrong with her I would try my best to ignore her begging for food. It is not the easiest thing to do but it is what I would do because in the end she is putting on weight that is not good for her and she knows now if she begs for food she will get it. They are really smart and know how to get what they want and sometimes we just have to ignore them for their own good.
It is great that you are looking for healthy snacks! Dog treats mostly are just empty calories that have to be included in a chis diet when considering how much food they are getting. I would rather give food rather than empty calories but I do give carrots and green beans as treats occasionally.
Good luck! I am sure you can get her on the right track!