How do I limit my message list to just show single contacts. Not show groups I sent to in the address list.

That could be for a couple of reasons. Are these SMS, or green bubble, actual text messages versus iMessges? While iMessages should not matter, and you should get them on both devices, especially the iPhone, you should check your settings for Continuity. See this support document. Use Continuity to connect your Mac, iPhone, iPad, iPod touch, and Apple Watch - Apple Support Check the section on SMS/MMS, including the part about Text Message forwarding. You might also want to check on your Send and Receive Settings, and make sure that iMessage is activated on both devices, and that Your phone number and Apple ID are checked on both devices. It could be that the iPad is getting the messages sent to the Apple ID.

Once the group appears in the search, you can tap on the ℹ and select remove to remove it from the list.