The paths are written by HTMLEditor using site relative. so if the root of your site is under the "SiteName" folder, then the url is correct, and it should not add the "/~MyUserName/SiteName" to the beginning. Can you give me some information on how to reproduce the issue on my end? If i'm able to reproduce the issue, i can assist you in getting the issue resolved.
Also, please tell me if you are on a mac or pc and the version of DW, and if you are using the Advanced Image Upload?