I don't know about riding in the car in particular but I can tell you that Lily was afraid of almost everything when I got her. It does take some time and patience to work through those things. Sounds like you're off to a good start.
At first, I couldn't even reach out and pet Lily without having her shy away. For the first 3 months or so I'd say, "Lily, can I pet you?" before reaching toward her. That got her attention and kept her from being startled. She soon learned what petting meant and got used to it. But I was only allowed to pet with one hand back then. If she saw both hands coming toward her, she'd bolt. I showed her that I was hiding one hand behind my back and reached only with the other. Lily was also scared to death of being covered with a blankie. She's tiny and often (very often) cold so a blankie is a part of our daily life... but in the beginning she would, again, bolt when she saw it coming. I'd wait until she was asleep and then place the tiniest corner of blankie over her bum. When she woke up she'd see it, sniff it and decide it was okay. I was able to cover her with more and more blankie until finally, she has come to love it.
If you start gradually and give Jasmine lots of rewards (petting, treats, whatever works for her), she should be fine. I wouldn't board her, given that she's so sensitive. That's exactly how I feel about boarding my Lily but at least I'd have a big advantage there - I'd be able to board her at the place she already goes for daycare. But to me, it wouldn't be much of a vacation without her, especially if I was constantly worried about her being without me at night. Although I'm sure it would be much tougher on me than on her. :lol:
Keep up whatever you're already doing and I really think it will work out well. Give Jasmine a big hug from me and Lily. :love3: