As far as losing weight without exercise, they say it's 80% what you eat and 20% exercise, so unless you have some type of medical imbalance, eating low amounts should cause you to still lose weight (though it may be a slow loss). Exercising will certainly help speed up the loss and can help sustain it (not to mention the more you are toned up, the smaller you look and the more calories you burn -- meaning you can eat more
) but you don't want to hurt yourself again -- I would suggest talking to your doctor, a trainer, or a physical therapist to get tips on exercises that you can do that won't aggravate your injury. Also, walk, walk, walk -- start with short walks and then lengthen them each week.
I agree that you probably aren't "hungry" after eating the fruit/veggies, so much as you are thirsty -- but I would say that snacks are always best if they are balanced, meaning don't eat fruit or veg alone, eat them with fat or with protein (or both, ideally). For instance, half an apple with a table spoon or two of organic peanut butter will keep you full much longer than a whole apple because it has fat and protein. If the veggies and hummus aren't filling you up, maybe you need to add some missing nutrients or more fat or more protein. I find that if I add a dash of olive oil and a sprinkle of feta to my hummus and then eat peppers, cucumbers and carrots, I end up feeling really full and satisfied (though it is higher call, it is also better macros).
Consider making your "meals" smaller, and having higher cal snacks or eating a couple small meals a day instead of meals and snacks. I eat avocado with a lot of my snacks/meals because it is creamy and high in fat, but still really healthy and a little bit goes a long way. I find that having meals that contain a bold flavor or that feel very rich help me feel more satisfied. One of my favorite meals is an avocado salad that I make by cutting an avocado into chunks and mixing it with some black beans, diced tomatoes, diced onions and a little bit of lemon or lime juice, salt and pepper and hot sauce (if I am in the mood for spice, I may also add some cayenne pepper). 1/3 -- 1/2 cup of the salad with some grilled chicken or grilled fish can be a really good small meal, and it's full of essential nutrients. To bulk it up, sometimes I add some quinoa to the mix or eat it with a few (8-12) tortilla chips like a salsa/guacamole.
Another thing, is make sure you measure everything. Often when people count calories they underestimate the amount of food they are eating (ie. two tablespoons can easily turn into 3-4 tablespoons of milk in your coffee if you are just pouring it from the carton). so you may think you are eating 1300 cals and then find out you are really eating 1600.
Remember, everyone is different and there is no one-size-fits-all diet, so find what works for you and your tastes. If it stops working, mix it up, figure out what you might be missing or what you might be overdoing.
Good luck!