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ok, first, stop babying her in the car. No coddling and petting and "oh it will be ok" That actually serves to make her anxious and make her think that there is something to be nervous about.
So, even if she isn't anxious, you can quickly make her that way. Constantly licking lips is actually a sign of nervousness. Does she do that before the drooling starts?
she is throwing up because her stomach is COMPLETELY empty, which by itself is enough to make many dogs throw up, even without the added car ride. That is why all the bile. If you are going to be going somewhere early in the morning, still give her at least a handful of kibble. Just a few bites to settle her stomach when she gets up in the morning. (Think pregnant lady eating crackers for morning sickness)
Work with short rides and get her accustomed to the motion of the car. Put her in the back seat or in a crate. Many dogs that get carsick are ok in a vari-kennel.
It's likely a combination of things. Puppy-ness that she will outgrow. Too much/too little on her stomach. Just plain motionsickness
Try the ginger snaps. Ignore her in the car, but keep an eye on her. When you see her start drooling, offer her a bite or 2 of the cookie. Ginger is a homeopathic remedy for carsickness. Short car rides. Seriously short - put her in the car and drive a mile. Stop have a play session. Get back in the car to go home. If she starts drooling sooner or is acting nervous, just drive around the block.
Watch for signs of nervousness. excessive licking (even drooling can be a sign of nerves, but can also be a sign up upset stomach). tensed body. how she is holding her ears.
Talk to your vet about carsickness meds if nothing else works. Even then, there is a chance she will eventually grow out of it.
ETA: I re-read your post. If she starts drooling the second she gets in the car, that IS likely nerves. Start off just getting in the car, turn it on and go to the end of the driveway, even if it's 20 feet. Park and lesson over for today. Again, no coddling or babying in the car. That tells her that she is right to be nervous and upset.