First of all, what camera is it? Then, how is it mounted?

If you have a Moultrie A5 (and what you describe sounds just like an A5), I can tell you that your problem is with the camera. I have two. They are practically plot-watcher cameras they trigger so easy. They do a good job of catching game, but get about 1000 extra pics as well. I can feed the pics into a video editor and make a stop-action animation movie of clouds going by during the day. At night, it does not trigger as often.

If mounted to a swaying tree, that can trigger the camera as well. If that is the case, pick a larger tree or use a T-post.

Cant the camera at a slight downward angle so as to not get so much in the way of tree limbs and sky.

You can try placing tape over the top half of the sensor (externally) to blind it from above motion such as tree limbs.

I would still try taping part of the sensor.

By chance is your camera set on a 5 minute delay?

What happens if you change to 1 minute? Will it then take every minute? Does it go off in a dark room with no motion? Easy test, put in closet for a while and see what happens.

Tried the 1 minute delay and it went every minute. I will try the closet idea and see what happens.