Cara,
I agree with the others, you are not fat! A bit "soft" maybe, but if we were talking horses, isn't that how we want mares to be?
Seriously, I'm 5'7" and 155 lbs, so I'm in pretty much the same shape as you. Of course, I'm also 47 years old and have had 2 kids, so I figure I'm doing pretty good! Yeah, I'd like to be a bit leaner, too, but let's keep this in perspective, shall we?
We keep saying "exercise, and eat healthy." What does that mean? Well, exercise can increase muscle mass, which in turn raises the rate at which you burn calories. Better muscle tone helps the way you look, too. Plus, the hormones released during exercise are natural mood lifters. Sadly, exercise by itself doesn't do much for weight loss, partly because, as you become more fit, you become more efficient in the way your body works!
So what about the "eat healthy" part? We all have our own definition of what healthy eating is. Here's mine.
Lots of veggies and fruits, as fresh as you can get them. Not only do they contain vitamins and minerals that your body needs, they also have water and fiber, which (among other things) help to fill you up with relatively few calories.
To me, "Vegan" is like a religion, so I'm not going to touch it. Unless you have some similar moral issues with it, don't skip the meat and dairy simply because of their fat content. At your age, you need tons of calcium, and it's hard to get it just from vegetable sources. Know what you're doing before you go crossing whole food groups off of your list! Protein and fat take longer to digest, so meals containing reasonable amounts of them will keep you feeling "not hungry" longer. Some fats are actually good for you, in moderation.
Carbohydrates are your body's natural fuel. They are what you burn best. Unless you have a metabolic disorder (like diabetes) you really shouldn't go "low carb." In fact, going extremely low carb can GIVE you a metabolic disorder! That said, don't overdo it. Simple carbs (like sugar) don't do you much good, and can make your blood sugar level act like a yo-yo. Get more "bang for your buck" by eating things like whole wheat rather than white bread, etc.
Generally speaking, anything with "artificial" in its name isn't really good for you. I call it "junk, pretending to be food." I don't avoid then like the plague, but I don't kid myself about them, either. Don't rule out any normal food entirely, or you'll feel deprived, and may just chuck the whole thing. My feeling about it is: if you really want the doughnut, eat the doughnut. But don't eat the whole box, and don't do it every day! A healthy attitude toward food should include a little room for being naughty. If you begin obsessing about everything you put in your mouth, you'll go nuts!
IMO, those are some general rules anyone can live with. As others have said, before beginning any diet or exercise regimen, check with your doctor. Take care, and know that we care! {{{{HUGS}}}}