If I understand your last post correctly, you are going to caulk the underside joint between the drip rail and the sidewall, which is only going to trap water behind the drip rail. The fact that rust is running down your wall means that water is getting through the screw holes in the gutter, and maybe all the way into your wall. If you stop the water from running out from behind the gutter, where will that water go?
As you've already been advised, I think the smart thing to do is remove and replace the screws, and caulk behind each one as you go. You can do them one at a time, so you don't have to remove the gutter. If you don't stop the water from getting behind the gutter, you are going to have bigger problems.
It is also possible that the builder didn't seal the top edge of the gutter, where the rubber roof tucks behind it. While you're up there, clean and caulk this joint as well. The Geocell caulk you mentioned should be a good choice for both jobs, just stay away from silicone caulk.
Finally, make sure your gutter is cleaned out, straight, and drains water. It looks as though it's trapping and holding water, judging from those rust stains at the screws.
Good luck!
Regan
