Is this case of weighted 2SAT NP-complete?
You can express the predicate "$x = y$" using one occurrence of each polarity:
$$
(x \lor \lnot y) \land (\lnot x \lor y).
$$
Consider now an instance of weighted 2SAT, in which each variable appears at most $M$ times. Duplicate each variable $M$ times, and enforce that all copies are the same using the gadget above. Replace each occurrence of each variable by a distinct copy of the variable. If the original instance asks for an assignment with at most $k$ positive variables, ask for at most $Mk$ positive variables. We obtain an instance of your problem which is equivalent to the original problem.

This shows that your problem is also NP-complete.