Is finding a solution of a satisfiability problem harder than deciding satisfiability?
As mentioned in a comment, any method of determining satisfiability of a Boolean formula can be easily converted into a method for finding the satisfying variable assignment.  This is because all NP-complete problems are downward self-reducible.

From Wikipedia:

Self-reducibility

The SAT problem is self-reducible, that is, each algorithm which correctly answers if an instance of SAT is solvable can be used to find a satisfying assignment. First, the question is asked on the given formula $Φ$. If the answer is "no", the formula is unsatisfiable. Otherwise, the question is asked on the partly instantiated formula $Φ$$\{x_1=TRUE\}$, i.e. $Φ$ with the first variable $x_1$ replaced by $TRUE$, and simplified accordingly. If the answer is "yes", then $x_1=TRUE$, otherwise $x_1=FALSE$. Values of other variables can be found subsequently in the same way. In total, $n+1$ runs of the algorithm are required, where $n$ is the number of distinct variables in $Φ$.