Will Elizabeth's unhealthy attachment continue on?  Will she still continue to see him as hers?  Possibly, very likely, even, but for how long, I don't know.  They're not going to really meet again for, what, three, four hundred years?  Elizabeth's active attachment is going to fade.  It has to.  Eventually, she'll go days without thinking about him.  But once they meet again - I can't say for sure.  I'd like to think that by the time they meet again, her obsession won't rekindle in the same way.  She will be more independent of him, and she will stay that way.  I don't know how jealous she'll get over seeing him with other lady vampires, I'm guessing that she won't get explicitly jealous or possessive unless she sees some possibility that he has feelings for one of them, or one of the girls do for him, if that makes sense.  For example, I don't know if she'd get possessive over Violet's relationship with Jaspin, because from how I'm seeing it, there's no romantic tension there.  Their relationship is clearly more like a master and his ward.

That being said, I don't know if perhaps Elizabeth will be more prone to interpreting any of Jaspin's relationships with woman as romantic.  That's a possibility.  I don't know anything for certain until we start playing with them in those contexts.  But as I foresee it, I'm pretty sure that by the time they meet again, Elizabeth will have enough control over herself and be over him enough to not go on some 'if I can't have him, nobody can' rampage over Jaspin just having women in his clan.

Right away would probably make more sense.  If he betrays her right away, it makes more sense that it's a knee-jerk reaction to this traumatic experience, before he can really get a hold of himself and think about what's really best.  For one thing, brand new vampires don't have a ton of self-control.  And two, if he had the time to cool down and get his head straight and really think about it, would he still really, with sound mind and body, choose to betray Elizabeth?