First you need to ascertain your ownership of the site/code. Check the contract and determine if you have full rights, or some other weaker rights. If you have full rights, you have the legal right to do what you will with the solution, whether that be transfer it to another provider, or whatever. Things can be murky if you have lesser rights, because the designer may own copyrights on the graphics, the design, and even the code behind any features that use code.
Second, regardless of the ownership issues, you should seek another web developer. It isn't really relevant why this relationship has gone sour, but obviously it has. Like all web developers. some are less service-oriented than others. Some things you probably should look for in a new developer are an insistence on a contract that defines delivery, ownership rights, etc. Also, you need to specify the due dates within reason, and provide some mechanism for change that doesn't injure your business OR the developer. Finally, you need an agreed document (doesn't have to be complex) that describes the site -- even with pictures -- so that everyone knows the targets.
I have seen countless situations where web developers take it on the chin because customers feel changes should be easy, not recognizing that while they may be technically easy they take time. It can become frustrating even for a good web developer if you don't have something in place to define how much the expectations will change over a given period of time. Having these kinds of understandings written down shows a respect for both sides of the equation, and that matters.
Good luck.
