Large gap between counter and backsplash

[birdographer]

I don't know if anyone will have any thoughts on this but I'll ask. We had a terrible contractor here to redo our kitchen. Part of the problem is that in places (one long stretch on one side of the kitchen, short bit in a corner of the other side) there is a large gap between the counter (laminate) and backsplash, 3/4" in some areas. He had planned to put down backing rods and then caulk over it. That wide an area of caulking, and only in a couple of places, does not appeal. [Note: I can't talk to him about this because he is no longer on the job.] We found that there were some recommendations to use short (top to bottom, not length) molding around the entire kitchen countertop, making it more uniform and covering the gap. While we can paint it to match the countertop and backsplash, we are concerned because one section of it goes behind the sink and faucet, an area that can get wet. We aren't sure what to do about this, never encountered it before (but then again, never had such an unskilled contractor before). Any ideas for what would be best? We can't afford anything major, we need to work with what we have been left with.