The main issue concerns who owns the content of the pages which are developed for your web site. If you own the content of your existing web site, then you can use that content in any way you wish. If you want to create a new site or switch content developers, then another developer can use your existing web pages as a starting point, saving on the time and cost of creating a new site as compared to having to start from scratch. If the content developer owns the content, then you probably will not be able to have any changes made to your web site except by the same developer. Graphical design talent varies widely.
Many web sites contain obvious spelling and grammatical errors.
In many cases, the developers own the content they have produced. There have been quite a number of cases where web site owners have discovered this only after they have tried to move their web site to another place. You should have the flexibility to make changes to your web site without restrictions. Therefore, make sure that you own the content of your web site, and that the web pages carry a copyright statement with your organization's name, not the developer's. When evaluating web site designers, look at other sites that they have developed, watching for spelling and grammatical errors.