"I am sure we will need at least one checked bag for all of us." What makes you sure of this?
Not knowing how large your family is or the ages of all the family members, each person should be able to manage their own luggage, including one major carry-on and one personal item.
You're only going to be gone about 2 weeks. You have access to laundry facilities. Your friends will have things like shampoo, conditioner and bath soap at the minimum. If not, you can buy all those things in Berlin and leave them behind for your friends if the containers aren't small enough for your 3-1-1 bags.
Speaking of 3-1-1 bags, with multiple people, you can spread your <=100ml liquid toiletries among all of you, making it possible to have everything you need without checking a bag with larger sized containers.
Typical carry-on size for international flights on major carriers is 22x14x9. If you don't take a major carrier, you may have issues with carry-on size and weight.
Since this is your first trip to Europe, it would probably be a good idea to read and absorb all the relevant advice on the RS Travel Tips. In addition to the packing light info there, it would be good to watch the video called Packing Light & Right.
The Explore Europe section on Germany would be helpful. And two good forums are on Packing and on Transportation.
I know this may be TMI when you asked about airfare, but all these considerations are intertwined.
One thing for sure is that you do not want to book your plane travel with some off-the-wall website that sounds like you will get a really cheap deal. Especially this close to your travel dates. That's the stuff that travel nightmares are made of. Like others have said, book directly with the airline.
It doesn't seem to matter how big the departure city or arrival city is, sometimes there is neither a nonstop option nor even a 1-stop one. You might have to go through somewhere in the States and maybe have an extra stop in Europe to get to Berlin from Austin. Pay attention to the total length of the flight, the carriers, where the plane changes will be and how long you have between them.
I messed around with Google Flights and saw nothing cheap that didn't involve a very long transit through multiple airports.
One option might be to fly nonstop RT between Austin and New York. Then on a separate ticket, nonstop RT New York - Berlin. It looks like there's enough time between flights to do that, but the cost is still about $1200 per person. And checking a bag with separate tickets is not a good thing.
For about the same price, you could fly Austin to London RT and RT London to Berlin.
BTW, Hook 'em!