Hey Natz,
I don't really know a lot about eating disorders but I do know about not being hungry

Several years ago when I was living on my own - money was really tight and I would have to skip a day without eating and days when I would eat would usually only be on meal...I found that the longer I went without eating - the more I didn't wanna eat when I actually could. I think my body just started "adjusting" to not being properly fed.
As to your question - your dizzy spells are from not eating. Because you are excersizing on an empty stomach.
Eating is not gonna make you fat - it's the OVER eating or eating unhealthy foods that will. If you are worried about 'getting fat' - try planning a menu for yourself - make them small portions of healthy food. And you already exercise so you'll be able to keep yourself in awesome shape without starving yourself.
You NEED to eat- your body needs food. Once you start getting back into eating - you'll find you actually want to eat. And eating healthy snacks inbetween meals will keep your hunger at bay. Your lack of appetite - in my opinion, is in your head. You've convinced yourself that you don't need food - you need to change that

Tell yourself that you won't get fat from eating this bowl of cereal in the morning - or that plate at suppertime.
If you still can't bring yourself to eat - talk to a nutritionist or your doctor - Don't let it go either - it can lead to serious problems - just search anorexia up on google. You may not be at that point YET - but if you continue like you're saying - you will.
Take care of yourself.
Sarah