Satisfying assignments, twice-3SAT NP complete
Hints:


Showing that the problem is NP is easy.
To show that it is NP-hard, show first that TWICE-SAT is NP-hard (see next step).
In order to show that TWICE-SAT is NP-hard, reduce from SAT. The idea is to add a "dummy" satisfying assignment, which is always going to be satisfying; any satisfying assignment of the original SAT instance will translate to a (different) satisfying assignment of the new instance.
Either reduce TWICE-SAT to TWICE-3SAT, or use insights from your proof in step 3 to prove that TWICE-3SAT is NP-hard.