It sounds like you are indeed frozen up. I would crank up the heat and hope that does it. If you have copper pipes I would recommend calling a plumber ASAP as many plumbers have an electric pipe thawing machine that can safely warm the pipes without the danger of fire. It is critical that you get this done as quick as possible to avoid burst pipes.
Once the plumber knows where the pipes have frozen you may be able to do something to prevent it from happening again.
I would not recommend allowing fixtures to trickle as the fact is running water can indeed freeze and the running water in many cases can overtax a septic system if you are on one or, cause the sewer line to freeze which is a lot harder than thawing above ground pipes.
If Niagara Falls can freeze like it did in 1911 rest assured moving water does indeed freeze.