I have the same issue (Sprint) - internet services are completely dark, apps wont load, texts/imessages wont go through unless I'm on wifi or I toggle the airplane mode. I contacted Apple Support and they told me to reset my network settings and to check with my carrier, Sprint customer care refreshed my network settings and told me to restart my device, once that didn't work they said it was an Apple issue and that there might of been an outage in my area ..
I went to the Apple store and after waiting for what seemed like an hour waiting to be seen by a doctor, an overwhelmed 20 year old kid at the Genius Bar reset my phone a few times then ended up replacing my device with a new one and said this was a Sprint PCS issue and that it could be a disconnect with the device SIM and the update. He also mentioned that this happens with Sprint customers every update, however this is the first time something like this has ever happened to me - you always hear stories like this and never expect them to happen to you. I got a new sim at a local sprint store .. and still after all this nothing .. a manager at the Sprint store said it probably wont be fixed until Apple releases a new revamped update that addresses all of these bugs. Save yourself some time and just use wifi until we get a new update !!!
iPhone 6s,iOS 9.3, Sprint