This contradicts something you said earlier, or perhaps I misunderstood.
So with 4g on, you can run speed test? That means your4g works fine. If you want another test, try using Pandora or any other app that accesses data. If everything is working, then it's likely a browser issue of some sort, and you probably don't need a new phone.
Google navigation is another app that uses data . Open it and scroll around while on 4g. The maps need to download as does traffic info. Turn on the traffic layer and scroll to the next city. It should have to download all that info. If that all works, then again, your 4g is fine.