This is what I believe...
I don't think Elizabeth has an altruistic bone in her body.
I do think she was toying with the idea of giving Gabriel up for adoption, in the beginning- but, even then, it wasn't so much because it would be the best thing for Gabriel, or to do something nice for a couple who couldn't have a baby of their own- no, she was toying with the idea of giving Gabriel up for adoption because she was lead to believe it would be easy, and because she liked the positive affirmations she was getting from other people (namely the Smiths). She liked the 'being commended' part of it.
After she arrived in San Antonio, it got too complicated for her. She became too frustrated. The whole 'adoption' thing was obviously not going to pan out the way she had imagined it might. She was told by (someone) on the phone that she needed to go back to Arizona to straighten everything out- that Logan was actively seeking custody of Gabriel.
Hell hath no fury like a woman scorned. This entire series of events was started after Logan broke up with her. I believe she had a feeling the break-up was coming long before it did. She was in a seething rage because of it, not for days now, but for weeks.
Remember, previously, when she destroyed the apartment (took a knife to the furniture, cut up Logan's clothes, broke mirrors, smashed all the windows, bent the frying pan, etc.) it was allegedly because Logan was late coming home from work.
Logan coming home late was nothing compared to Logan ending their relationship.
How devastating that must have felt to Elizabeth, who has two dead parents - a drug addicted father, and an alcoholic mother- and who had been placed in five different foster homes.
It is easy to guess that Elizabeth probably suffered not only from attachment issues, but also from abandonment issues, and it's not too much of a stretch to believe that Logan breaking up with her was yet another blow to her emotional security.
Her hurt would have run deep. And, not having the emotional maturity or the coping skills to deal with this new 'abandonment', the emotional pain would have quickly manifested itself into rage. Rage, she is familiar with.
To then learn that not only was she not going to get Logan back, but that he also wanted to take the baby 'from her' (the baby she had already been threatening to put up for adoption to get his attention)- that would feel like another 'abandonment'.
There really is a thin line between love and hate.
~I could go on, but this is already getting too long. I believe she killed Gabriel, because she wanted Logan to hurt as much as she herself was hurting.