1200 calories is not a lot, but it depends on 1200 calories OF WHAT FOOD. Are you eating protein and veggies, or are you eating cakes, sweets, and chips? You will not feel restriction if you are eating sugary or salty snacks because they dissolve into nothing, but if you eat lean protein, you will likely feel restriction. For example, I went out to sushi last night, but didn't eat the rice, so I essentially just ate raw fish - pure protein. I was stuffed! But then after I got back to my hotel I found a rice krispy treat and was able to eat half of it with no issues... oh well.
I also have been hungry since day one! I was put on metformin and topamax (topirimate) by my medical team, and it definitely helps the hunger. So does tea and coffee, and drinking enough in general.
Also, two weeks of not losing weight is totally normal.
As far as exercise... gentle walking likely won't make you hungry (except if you're "expecting" it to). It is good for your mood and while it is unlikely to help you lose weight (you can't out-exercise a bad diet!), it is excellent for health and it is good for keeping weight off long term.
Good luck!