I have flown with window cards many times. I always just carry them in the plastic bag they come in. A few times, I have had flight attendants offer (without me even asking) if I would like it placed in the first class closet (without sitting in first class). Other times, I either just place it underneath the seat in front of me so I can keep an eye on it the whole time, place it in the bin above me after everyone has seated, or if I'm in a window seat, I will just slide it in between the seat and the window. It's the perfect fit, it stays safe and secure and doesn't bother me or anyone else.
I have done all of the above multiple times and have never damaged or even slightly damaged a window card. Just be mindful, have a few plans about what to do with them, and you'll be fine.