This is doable with the seats in place...just move them as forward as possible for easy access to the rear panel(s).
As you said, two screws hold the panel in place. The panel has a vertical lip on each side that fits along the same vertical groove along the seat edges. The seatback springs provide the tension needed to hold the assembly in place...so pressing down on the seat from the front will alleviate the load on the panel and make it easier to remove. It's not obvious at first, but once you've done this, you know how to go about it the next time.
Just remove one edge. You can swing the panel outward enough to gain access to the internals.
The headrest assembly is $16.50 per seat. I bought two for insurance.
Replacing the assembly is about a five-minute job. Interesting you say you can't move your headrests "up", as the usual problem is just the opposite. The plastic "clamp" fails, therefore no grip is provided to pull the headrest down, but there is usually sufficient material left to push the headrest up by inertia. I guess the clamp has self-destructed in your case.
Hope this helps...