I understand that you are having some issues with your Slingbox PRO-HD ever since you switched to a new ISP.
Every time you connect the Slingbox to a different router, it will pick up a new IP address. This process will allow the communication between the Slingbox and the Internet. Now, this also means that you will need to reset the Slingbox in order to clear out all the "old" information.
The Slingbox reset process is explained through this article:
Once your Slingbox gets solid lights, you will be able to run the Setup Assistant for Internet Viewing. I encourage you to run the Manual Configuration. Further details are included on this link:
Since you have not included any information about the make and model for your router, I recommend you to use the following link in order to get the step-by-step instructions that will allow you to set up your Slingbox for Internet Viewing:
Hope this helps!