Your HR44 Genie is a SWM-only device. Your old dish uses the older "multi-switch" system. The simplest thing to do is to convert your existing dish and LNB (assuming your LNB has two coax outputs) to SWM by adding a SWM8 switch and a PI-29 power inserter. You can get both of these on Amazon but I'm not sure if they would be available in a local store. Any DirecTV installer would have them if they will sell to you.
To be very clear, if your LNB has two coax outputs you don't need to change your LNB just add the SWM8 and the PI29. If you have only one coax from the LNB you would need to change to a SWM3 LNB; in that case you would need a PI-21 power inserter and wouldn't need the SWM8
If you aren't familiar with setting up a SWM system, take a look at this guide posted by Winegard. You don't need to have a Winegard system to understand how the wiring should be set up; just use the basic principles.
Using non-SWM hardware in a SWM system