A new service was added after version 10.4.3 - I don't know if it's in 10.4.4 or doesn't show until version 11 - I do know it wasn't there when we tested with 10.4.3, but was there with 11.
In the C:\Program Files\Laserfiche\Laserfiche Forms\Forms\Web.config file the line looks like this:
<endpoint address="net.tcp://localhost:8172/lfinstance" binding="netTcpBinding" bindingConfiguration="timeoutBinding" contract="Laserfiche.Forms.Routing.IInstanceProcessing" name="" />
In the C:\Program Files\Laserfiche\Laserfiche Forms\Forms\bin\RoutingEngineServiceHost.exe.config file there were two lines that looked like this:
<add baseAddress="net.tcp://localhost:8172/lfinstance"/>
and
<endpoint address="net.tcp://localhost:8172/lfinstance" binding="netTcpBinding" bindingConfiguration="timeoutBinding" contract="Laserfiche.Forms.Routing.IInstanceProcessing" name=""/>
As you can see, those three points all reference port 8172.
When running this line from an administrator elevated command prompt:
netstat -ano -p tcp
We were able to see that the port was being used by a different PID than Forms - in this case, one that showed "System" when we checked the Details under Task Manager.
Under Windows Firewall, we found rules indicating that the port was being opened for use by "Web Management Service". A little more research, and we found that is a service used by IIS in order to allow the Web Deploy functionality (and probably other stuff too). I had installed Web Management Service and Web Deploy because I have several custom sites that I have created and publish from Visual Studio directly to IIS the server. This hasn't been an issue on the older versions of forms, but with the new version trying to use port 8172, we couldn't get any forms to route through the system properly.
This port conflict was the source of our issue.
We switched the three references in the configurations to use Port 8176 instead of 8172, added Windows Firewall inbound and outbound rules to allow access within our domain to port 8176, and rebooted the server. After that, all the errors were resolved, and Forms began operating as expected.
Because this was such as weird situation, I wanted to share the solution here in case it helps anyone in the future.