Your goal is not to cut down the communications. It is a symptom of a lack of a process to escalate problems, which is amazing that you went live without it.
You did not say what the project or tool is but a problem management process is quite extensive, is not cheap, and you can get a lot wrong if you don't build it well. You have to design the work flow, the controls, the service levels, how you will measure performance...all while keeping those costs at a minimum. All of this should have been done long before the project went live.
If you try to control comms now without a viable alternative IN PLACE, you risk creating other, more severe issues for the company and its work. This is a "fix one thing, break two others rule. For example, trying to have the communications flow through a single source, especially if that source filters, you risk a bottleneck, inappropriate filtering, bypassing, and most importantly exacerbated user rejection.
So do not focus too much on the symptoms. Stop the bleeding, but focus on the source.