I totally agree, this happens when you first start. Call it inexperience or just plan ignorance. Either way if this designer is any good they will have new work in no time. Learn to manage your clients better and how to actually approach a project correctly. On another note, take them to court if you think it's worth the time. Or if you can take them out to lunch and figure it out with out legal council. See if you can make a new friend, it's better to have them as a friend than an enemy.
Whoa $780 for that design? I personally would not have even charged for that preliminary design. Why are you designing any way. Where are the rough sketches and wireframes? Did you not learn that in school? Wireframes first, then discuss with management to get a better understanding of the underlying goals and objectives. Get them on the phone and ask lots of questions about the company, management, customers and the overall scope of the project.
Looking at that design makes me think the designer is fresh out of school. Lack of a real understanding for client management and asking the client real hard questions about strategy and long term goals. A bunch of whiners, get over it learn to handle your clients better.
I wouldn't pay you for that design. It's mediocre and needs major refinement. I would tell you to redo it. Get over it and learn to hustle better, Otherwise you won't make it very long as a designer. Verbal agreements don't mean anything any more. Clients just don't care and designers are a dime a dozen these days. If we fail then our competitors pick up our mess. Again I will say, get over it and find some other work. If you are as good as you think you are then you should have no problem finding work, right?
