I honestly haven't ever used the shower/tub for it's usual purpose. We use campground facilities for that, or a lake, or river .... We use the tub for laundry. I do think mine was installed right though (luckily).
Shimming it up doesn't seem unreasonable to me. You'd want to make sure the shims wouldn't be able to vibrate out of place while the trailer is going down the road, so likely would want to put a little glue on them. If you re-caulk, remove the old caulk first.
DIY? That's really up to you. I know this guy that thought replacing the axle was a DIY project ... but wouldn't do it again!
