Restriction of SAT to CNF
You can apply Davis-Putnam procedure and note that with the given restriction applying resolution rule will not be increasing formula size exponentially (in fact, it will be decreasing it). Thus, every iteration of procedure will have polynomial time complexity.
Read about Tseytin transformation and try to exploit the same idea. You can replace variable $x$ with $y$ and add $x \leftrightarrow y$ to the formula. With a proper replacement strategy you will be able to obtain an equisatisfiable formula in which every variable occurs at most three times: one in the original formula, one as the left part of equivalence, one as the right one.