The line is cheesy, and not in a good way. But, the sentiment is a good one. Have you never been in love with someone so much that the mere thought of them hurts because you're not together? Have you ever loved someone, been close to that person on a regular basis, but know that you can't be with him or her? That's what that line means. Anakin was so in love (some detractors might say merely infatuated) with Padme that the fact that he was not with her caused him anguish.
And therein lies one of Anakin's flaws, and it's not a flaw of his character, but his upbringing. I don't believe the Jedi Order really prepared most kids with how to deal with emotions well. Certainly, in Anakin's case, he was poorly equipped for them. Love, anger, fear ... whatever the emotion, Anakin lacked the tools to deal with them in positive ways. And that more than any other thing was what I think fated his downfall.
As to the OP's question regarding Padme herself, I like to think that Padme is the idealist she's portrayed as being. But, I think she also knew that telling anyone what happened on Tatooine wouldn't do much good. No one on Tatooine likes the Sand People. Hell, they'd probably give him a medal. The Lars family certainly would celebrate what happened. On Coruscant, all it would serve to do is bring Anakin's future as a Jedi into question. Better to keep quiet and let him learn the lesson on his own.
Of course, to use the vernacular, all that did was enable him. Perhaps if she had told Obi-Wan, then he could have talked to Anakin about those feelings. It's not as if the Jedi are unforgiving to people who backslide a bit. As far as keeping the marriage a secret, that was probably completely necessary. The Jedi forbid attachments, and Anakin surely would have been kicked out. It wouldn't just be for being married, as a Jedi could apply for a special dispensation for marriage. But, he married a Senator--a politician; that would, I think, be the greater problem.
You make a good point.
However: you say many of Anakin's flaws have to do with his upbringing (essiantially the Order's fault). I strongly disagree. I remember reading a book about when Anakin was a young padawan, and compared to the other padawans (who were also brought up by the Order) Anakin's character was still bad.
That kid was just messed up. If I had to use one word to sum him up, it would be selfish.
Sorry. This has been off topic.