iphone 5 - cellular data not on ??

I'm a bit confused about this one. I think this started happening yesterday. I sometimes turn my WiFi off to use just cellular, especially while at work, where they restrict which websites you can visit. When I am at my desk, I usually have a strong cellular connection (Verizon) - full bars, LTE. What's odd is, my Safari browsing is not working at all. It tells me I am not connected to the internet or a pop-up message will appear telling me to that my cellular data is off, while also recommending I turn on my WiFi. But when I go to settings, my cellular data is of course set to On. The odd part is, my Facebook app works fine and updates fine, my Mail comes in regularly and works fine. Everything that needs data is working, except my Safari. And the whole time, I have what appears to be a strong signal.

I have tried restarting my phone, turning my phone off and on, closing Safari (and all my other apps), clearing my Safari history, clearing my Safari cookies and data. I'm running out of ideas. I tried turning the cellular off then back on. Nothing is working to get my Safari to work.

Any ideas?

(As a sidenote, I get a similar message when I am on the road and I am using Maps - the phone wants me to turn on my WiFi even though I am no where near my WiFi networks. Why would the phone need the WiFi turned on if it's only able to use Cellular?)