Thanks for the additional information. It is helpful.
It does sound like separation anxiety. Given the severity of her response, I would do training at home first and then work on the problem away from home. I would start with a hard plastic sided crate for her. Put her in it at home and leave the room. If she is calm and doesn't react, go to her immediately and reward her with a hot dog treat. If she objects try moving closer but don't let her out until she stops complaining. When she does, give her a treat and let her out. Repeat gradually moving further away from her until you are out of sight and she isn't carrying on.
I suggest a crate inside so you can transfer the crate that she knows and her behavior in that crate to the car. Remember to give her tiny hot dog treats when she is calm in the crate. Next transfer the crate to the car and put her in the crate. Don't move the car but take yourself a little away from the car. Start again by only walking a short distance away. Perhaps even leaving the door open but the crate in the door. Go back and reward the desired calm behavior. Eventually shut the door and walk away. You will be doing this gradually so she learns that even though you leave her in the crate, you come back and reward her with calm praise and tasty treats. . You will gradually work up to her being out of your sight. Now if she does carry on, you want to ignore her until she settles down which is why you do it at your home where you can let your neighbors know that you are working on training her not to cry like that. When she settles down, the the praise and treats and even letting her out. Repeat as necessary until she is no longer reacting to you being out of her sight when you are at home and she is in the car. Then you will want to move to another location away from home but still isolated for the next phase of training where you will repeat the training exercise.
Soon you should be able to take her anywhere and leave her for short periods of time without her behavior being a problem. Now a dap collar might be a help as well. They use pheromones to calm a dog and are proven to work for separation anxiety. Your vet might also suggest something like xanax as well but you do need your vet to prescribe that. Medications can be helpful when starting training but th dog needs to come off them eventually. Dap collars can be bought online, in pet stores and at your vet office usually.
I never recommend another dog to solve a separation anxiety issue. You may end up with two dogs displaying the same behavior.
I hope this information is helpful to you. If you would like any additional information or have more questions please don’t hesitate to press the reply to expert or continue conversation button so I can address any issues you still have . If you do find this helpful, please take this opportunity to rate my answer positively so I am compensated for my time.