Hey Mina,
I'd ask a similar question -- how do you know when children are hungry? Well, they get whiney and lethargic and ill-tempered. The same goes for me -- even aside from normal hunger pangs and feeling woozy, I can usually see my behavior change after I'm hungry. I think most adults are the same; we just hide it better than children (which is why I asked about children).
As for feeling full -- well, that's just when you're full. Some people just don't feel it very well. The one thing that helps is to eat slowly. I usually eat very slowly, because I'm occupied with something else -- either conversation with whoever I'm eating with, or doing a crossword, reading, or something like that if I'm alone.
As for not calorie-counting, I'd suggest portion sizing as an alternative. I'm a bit the opposite of you with vaguely similar results -- I can certainly tell when I'm hungry or full; it's just that I don't really care. I can be ravenously hungry for a long time, and just ignore it, not out of any eating disorder, but because I just don't particularly care -- laziness, basically. But if I have food I like, I can just go on eating it. I'll tear through a bag of potato chips in half an hour. Baked goods diappear quickly.
So I get proper nutrition by keeping my as diet diverse as I can, and having a standard portion -- a medium bowl
of cereal or a small plate of eggs, toast, etc. for breakfast (usually two eggs is enough for me -- as opposed the French, for whom one egg is un oeuf

); for dinner, a full but not overflowing plate, a sandwich for lunch, only one drink most of the time in the evening. Sometimes coffee and a patry in the afternoon if I am feeling down.
Anyway, the point is, the calories you get in any one day alone don't matter for maintaining a healthy diet -- it's the long-term average. So you can do a rough calorie count ahead of time by knowing what portion sizes are right.
And here's the key: I prepare at least part of what I eat when I eat it. So if I'm having rice, I make just enough of it for that evening; everything else I serve in proportion to it. If I prepared more, I'd have seconds -- because it's there. But once the rice is gone, well, I'm done. If I'm really hungry, I'll make something else, but that takes more effort, so I don't unless I really need it.
Anyway, that's how I do it. That and regular
fun exercise -- hiking, skiing, pickup games of frisbee with friends, at least once a week. Hope this helps. Good luck, and best wishes!
~Alyssa
p.s. This doesn't work when you're in grad school.