Is embedding a solution feasible for SAT?
You can take any formula $\varphi$ and change it to the formula $\varphi \vee \psi_x$ where $\psi_x$ is a "hard" SAT instance whose only solution is $x$. One way to construct such a formula is using cryptography:  if $f:\{0,1\}^n\rightarrow \{0,1\}^n$ is a one-way permutation and we choose $x$ at random and set $y=f(x)$, then one can convert $y$ into a SAT formula such that $x$ is its only solution and thus finding $x$ corresponds to inverting $f$. (We need for this $x$ to be random, but this something similar is anyway assumed if we think finding $x$ should be hard.)