Take a look at
this thread made on another forum by one of your states very helpful members - John Barca. It describes, complete with many pictures, how he sealed up his camper with Eternabond tape - including the gutters. I think his post will help you a lot.
He is still active on our forums and used to have a post like the one I linked but Photobucket took his pictures down so the pics wouldn't show anymore. He will probably eventually get around to putting that post up here. :)
Edit: I would do #2. In addition, I would remove each screw, squirt a bit of caulk into the hole and replace the screw - then cover with the Eternabond.
Another possible option would be to install
RainKap in place of your existing gutters. It installs in place of the existing vinyl strip. I think this is what TurnThePage was talking about a couple of posts above. Looks like this product is very hard to find now and must be out of production.
Barney