Is it normal to be obsessed with eating vegetables?

Binge eating is certainly common, and it's pretty unhealthy, but if you're gonna binge on something fruit and vegetables is the way to go. I don't know how much vege binging is considered constant to you, but I'm going to assume that this is a case of "I'm binge eating and the food happens to only be fruit and vegetables" as opposed to just "I'm eating significantly more fruit and vegetables than my peers and my family doesn't respect my dietary choices", which would be more of an issue of you needing to work it out with your parents than anything else, as most people don't eat enough fruit and vegetables.

If you tend to eat more of them when you are bored or have nothing to do with your hands (like when watching TV or videos) it might be a case of an oral fixation. That's pretty common. I have one myself, but I usually go for nail biting or eating significantly less healthy foods. One thing I've started doing recently is keeping a bottle of water on me at all times. From doing so I've developed a habit of drinking water instead of eating or chewing things. It's a double positive of doing less chewing and also drinking more fluids.

If you tend to eat more fruits and veggies when you are feeling stressed, sad, uncomfortable, or some other form of bad, it could be a case of comfort eating. I still would recommend trying out the water bottle, but it's less likely to work in this case (it's still a good habit though so why not). If your problem is a form of comfort eating, you should try and catch yourself when you are going to the fridge or the fruit bowl and address the actual problem, the discomfort, instead.
If it's something you can do something about (like stress because you've been procrastinating on something, loneliness because you haven't hung out with your friends for a while, that sort of thing), do what you need to do to make yourself feel better. Sometimes it might be hard to get motivated, but if you push yourself to do what you've been neglecting, you'll feel a lot better when you're done. If the issue is out of your control, you need to mentally address this and accept it. If that's not enough to make you feel better, distracting yourself with work or a hobby is the way to go. It doesn't need to be productive, so long as you are doing something besides stewing on bad thoughts and/or eating more food than you can afford.

If you find that you are often feeling hungry (needing food as opposed to wanting it) or getting low on energy despite eating so many vegetables, you might have a deficiency in something like iron or protein. Both can be obtained from meat (esp. red meat). I don't know how much meat you eat or if you eat any, but I know that meat is even more expensive than vegetables. Leafy greens like kale and silver beet are a good source of iron (the darker green the better), while nuts and seeds can be a good source of protein (but nuts can be very expensive). Making a smoothie for breakfast can be a great way to get a good variety of fruits and vegetables into your system in one sitting, and can help you get a more balanced vitamin intake. They also tend to be really energizing, and help you last better if you have a long day of work or school ahead of you.

I wish I could help you on the financial side of things, but I get the impression that you don't have much money available to play around with. If your issue sounds more like my third suggestion, I would definitely encourage discussing this with your parents and possibly talking to your GP (if you can afford to at some point), because if you're overeating because your body is deprived of something, varying your diet could possibly even lead to money being saved on redundant foods.

Are you serious in saying you can't afford fresh vegetables except carrots? Your wanting them sounds healthy to me so if I were you I'd be doing something about reorganising your finances so you can afford to buy at least more than just carrots.

Good veggies huh... Eat em all and drink a lt of water...
Water is the key... Always remember that...