Dear J.,
Congratulations on your baby!
Have you tried putting your daughter in her car seat in the house? Just buckle her in with a little music on while you fold the laundry or something and see how she does. It might give you an idea as to whether or not it's the car seat or the motion of the car that she doesn't like. It might sound strange, but I've known babies that actually slept very well in their car seats, in the house. Maybe they felt secure, I don't know. I had a car seat that popped in and out of the car and if my son was asleep when we got home, I just carried the whole thing in and let him sleep.
If she does fine in the car seat in the house, it's likely the motion of the car that's upsetting her. If she still fusses in the car seat in the house, you might try a different one so see if she's more comfortable.
She's only 10 weeks old and she may not like being in the carseat OR the car, but it's obviously something she'll have to get used to.
It's so hard to hear our babies cry and not be sure what to do. But sometimes they just have to cry and moms have to get used to it.
My daughter was 3 days old on our first excursion. My milk came in. I didn't own anything big enough to wear that would fit around my chest (thank God for dad's shirts!). He was at work and I was in desperate need of a breast pump, or something. My baby and I headed out to find the nearest drug store. She was screaming, my boobs were ready to explode and I was sweating like I'd run a 10k. That was just getting out of the driveway.
I know how nerve wracking it is when your child is crying and you're trying to drive. I would definitely try a pacifier for in the car, I don't see what it would hurt and if it soothes her, it will make you less nervous and your outings will be safer. It may turn out that she just doesn't like being in the car and for now, all she knows to do is cry about it. You've been great about trying just about everything under the sun so if a different car seat doesn't do the trick, she just may not enjoy the car until she can face forward. Some babies don't like baths either, but they still have to get washed. And you still have to get from Point A to Point B on occasion, so there may be some car crying in your future.
When I was a little kid, there were no seat belt or car seat laws. I shudder to think of all the long trips we took and my mom made a "bed" on the floorboard of the car for my little sister or just held her. Little ones don't understand that car seats equal safety and it's not hurting them to be in one.
I hope you find something that works for you and I'm sure you'll see the day your daughter can enjoy getting to go in the car with you.
Best wishes!