Translation:We did not stay long. We left right after dinner.
"right after dinner" is the expression used. We do not use "the" in this case.
Still, it could be correct in some contexts, like if "the dinner" is a particular event. As part of a conference, for example.
"right away" is good if we are not saying "after" something, here it is "right after dinner"
"Right (away) after (the) dinner" – the versions with "away" and/or "the" are all idiomatic (natural-sounding), with the various possibilities potentially carrying slightly different nuances, though all of them have effectively the same meaning.
In French apparently it's also possible to say "juste après le dîner", though perhaps it's less common. Native French speakers?
There is a problem with the set up of the clicking of the boxes containing the words. I was not able to click on "dîner". It gets buried underneath the words that are already selected. I had to click on can't listen now in order to proceed. I hope this is fixed.