New dog terrified of walks and car rides

Hello,
I adopted a 7 month old Lab/Beagle mix a week ago. She needs lots of exercise, but in the last week I have been unable to get her leave my tiny yard.

Today, I tried a new approach, I picked her up, carried her to the car, and took her to a large fenced-in park. She was quite scared during the short ride, and very scared of getting into the car, but LOVED running off leash in the park. We ran together for about 20 minutes before another terrifying 5 minute ride home.

So, right now I am not quite sure what to do. I see a couple options:
1. Keep taking her for rides to the park, despite the fear of the car, giving her lots of treats once in the car to help create a new association.
2. Continue helping her get used to walking on a leash and hope that in time she'll be able to walk past the end of the driveway.

It is tough to tell if she will get over these things on her own with baby steps, or if she needs to be pushed outside of her comfort zone with lots of love and support to make progress.

What I've tried:
I have followed the advice of Zak George and others using clicker and treat training to get her comfortable with the basic commands - she is already good at sitting, laying down, and coming on command. She is willing to walk on the leash back and forth in my small yard (50 feet across). However, as previously mentioned, she has a strong fear of leaving the yard.

Considerations:
- When she is not properly exercised, she is destructive in the house, and has a tough time relaxing inside. However, today she was properly exercised and has been exceptionally happy, well behaved, and relaxed in the house. (Is this worth the morning stress? Am I making her car anxiety worse?)
- She is headed to the vet next week for the new dog checkup, so I am hoping to get her some positive car associations before then

Any ideas or advice would be much appreciated! Sorry for the long post!!
-Will