It's not strength. It's not fortitude. It's self-preservation. Dawn did not and does not want to die and she knows that handing herself over to the bad guy will likely result in her death. I honestly don't think anything else occurred to Dawn.
When Dawn was standing on the stairs, eavesdropping on the conversation between Buffy, Giles and Joyce in 'Triangle', she didn't run away then. She just heard something that made her go 'huh' and start to question somethings.
Dawn doesn't actually run away and find out the truth until 'Blood Ties' - TWO episodes later, two episodes that translates into two weeks in-show time, at the very least.
In the time, Dawn has started to realize that things aren't adding up, that people are talking about her because they stop talking whenever she enters a room. She doesn't actually suspect that she's anything mystical or magical or important. When she runs away from Buffy's party, she's only doing it to piss them all off more than she's hoping to actually find something out about herself.
And when she DOES find out, she runs straight back to Buffy, because that is where she knows she's safest. Even when they're fighting, Dawn will always run to Buffy first.