I'm so sorry for the stress/frustration you are feeling during this time. I don't have kids so I can't even pretend to say I understand what you are feeling, but I do just want to extend a hug
Exercise is important for your health for so many reasons but if you can't include regular exercise in your day due to your baby, weather, etc, then you should still be able to lose weight. The saying is that weight loss is 80% diet, 20% exercise. Some lose without any exercise. Not to say you shouldn't try to get in exercise, but I don't want you to feel like the whole thing is hopeless if you don't have a chance to go run or join a gym right now. You *can* do this.
Health food doesn't have to be the super expensive protein-flax-omega3-fancy stuff at Whole Foods. Simple steps at first- cutting out chips, sodas, candy, and fast food will leave you more money to spend on fruits, vegetables, and lean proteins. Whole wheat bread is pretty similar in cost to white bread, same with brown versus white rice and whole wheat noodles instead of regular ones.
It sounds like you are at home a good deal, which perhaps means you have more time than money right now, which is perfect for weight loss. Spending an hour or so every other day cooking homemade easy meals with healthy ingredients will cut down on the need to eat out/order in (if you do that), also saving money and increasing your intake of nutritionally healthy food. Homemade spaghetti sauce, chili, and lasagna are a few very easy recipes that just require 10 or so minutes of prep and then 1 minute to take out of the oven/off the stove. If you like cooking, you could even start experimenting with making your own breads, snacks, etc. When I have time I love making homemade pita chips, which is just flour and water- super super cheap. Added bonus: standing on your feet cooking burns calories
There are a lot of exercises you can do in the house with minimal equipment. Check out the Exercise forum for ideas- I think there is a sticky thread at the top about exercises to do at home (or just search for one). Lunges, pushups, planks, deadlifts, etc are all easy things you can do to get the blood flowing, work your muscles, and raise seratonin levels (which increases happiness

)
Good luck and hope you can find the happiness and love you deserve!