6 hours in TXL and 5 in DUS.
I don't know anything about DUS, but about the TXL stopover, that should be enough time if you wanted to go to shul to daven etc.
Otherwise, Re. sightseeing, if you've never been to Berlin, I'd say to take a train(s) (to get to the U Bahn (subway) from the airport, you have to first take a short ride on a city bus) to Brandeburger Tor (name of station) which is the famous Brandenburg gate. Once you're there (and done snapping photos), if you turn a minute in one direction, you have the Bundestag (parliament), in the other direction is the (straight ahead) is another famous monument -Tiergarten (similar to 'the national mall' in DC), and in the other direction is the holocaust monument (and museum under it) - open 24hrs.
At the Brandenburg Tor, there is a tourist center, that can give you lots more interesting spots to see, all around this very historic area, there are many more interesting places and museum.
If you continue a few blocks down, in the direction in which you turned to get to the shoah monument, you will get to Potsdamer Platz, the financial center of Berlin, also a stop on the U Bahn and the S Bahn.
There are so many more places to see (e.g. the Jewish museum, topography of terror, Sachsenhausen concentration camp, and so much more), but there's no chance you'll have enough time to see them.
Bon voyage!
