Getting image urls to match custom site url

I’m having some problems with the URL for my website. The wordpress install is in a subfolder of a larger site (let’s say http://www.example.com/mysite), but it’s the only wordpress install, so this isn’t a multisite. The problem is that we don’t want it outwardly associated with example.com, so we used the Site URL option in the general settings to use a custom URL (http://www.mysite.com). It works fine for links, but image URLs are still displaying the original URL (http://www.example.com/mysite/image.png). Also, http://www.mysite.com/wp-admin redirects to http://www.example.com/mysite/wp-admin, which isn’t ideal either. Does anyone know if it’s possible to at least fix the image URLs? Thanks!