For these network issues that you are experiencing after changing the router, we recommend resetting the Slingbox. Please have a look at this link, there you will find the instructions and after doing this you just need to run the Setup at Slingbox.com:
Trouble watching locally or over the internet? Open up your port (port forwarding to your slingbox) on your new router and it should work.
One issue I had with my sling is that I had to force my router to assign it 192.168.0.254 via DHCP. Seemed like it was always resetting its static IP to 254. No kidding, my router assigns it .108 and then a day or two later I could no longer connect. I find it is listening on .254. ....WTF
So I just set it to always assign .254 and that got rid of the problem.
I am not sure resetting your slingbox will make any difference at all. It doesn't care about your router, as long as it gets an IP address, can talk to other devices on your network and has internet connections being forwarded to it.