Our payload has no stabilization either, and in past launches we've had quite a bit of rotation and swinging. Here's a video of one of last launch, for reference: https://peertube.scd31.com/w/7CQCYB4BmJzngoZTiMociY

To point the camera we have a lot of clever software and hardware. I didn't have any part in it, so I'll try to explain it as best I can. There's a diagram here, which hopefully will help you to follow along: https://eclipseplus.ca/Project_Details/Payload_Design/

We have a fixed camera mounted above a mirror. The mirror is on a special gimbal, which can be maneuvered using two servos and a bit of math. For coarse aiming, we have an IMU on board which uses the magnetic field of the Earth to figure out its orientation. For fine aiming, we do a bit of image processing to try and center the sun in the field of view of the camera. It's not perfect, but since we're only capturing still images, it's okay if not all the images have the sun in view.

Also of note is the filter, which starts in-place (to protect the optics) and automatically moves out of the way during totality. It also moves back into place afterwards so that we can continue safely taking images after the eclipse.

The balloons on the landing page are accurate! Those pictures are from past launches, which used the same payload (with some changes between each launch)