A vegan diet might help you, but most people with bad breath find that their surface level bad breath goes away while there is still an odor radiating from the back of their throat. That is because when you do not eat any kind of animal protein, all that bacteria that would normally rot in your mouth and cause an odor is not there. However, if you eat too many fruits or starches, then you'll still have a bacterial overgrowth in your small intestine which will cause an odor from inside your body rather than your mouth.
You're on the right track though, leafy greens will definitely help you as they're a natural source of chlorophyll.
See if a vegan diet helps you, remember focus more on the green veggies. If you find that it does completely solve the problem, then that's great! However, if it does not, then still focus on greens, but remove all fruit and starch from your diet and add in lean meats and seafood. Still, focus mostly on green veggies and remember that it's okay to eat just green veggies as a meal sometimes. Don't eat eggs, they are very sulfuric and terrible for any Body Odor .
Also, don't bother with commercial mouthwashes or toothpastes, even the ones marketed towards people with chronic halitosis. Gargle a mixture of water and activated charcoal. It absorbs odors. If that only helps a little then drink it too. Also, *drink* liquid chlorophyll. Don't bother with mouthwashes, they're bandaids, they don't solve the problem. Only diet and careful supplementation does, but the supplementation only works alongside a diet. Unfortunately there will never be an easy way out.