Critical properties are studied in systems with quenched bond disorder that is correlated along d1 of d dimensions. A renormalization-group scheme (based on the Migdal-Kadanoff method) which follows the full distribution of the random bonds and which gives correctly the modified Harris criterion +d1 is used. For d1<d-1, we find fixed distributions at finite temperatures, yielding new random exponents for various q-state Potts models. For d1=d-1 there is no long-range order if there is a finite weight to zero coupling. Otherwise, we find a novel zero-temperature fixed distribution, for which all the moments diverge to infinity with finite ratios among them. This fixed distribution has a magnetic eigenvalue equal to d, indicating a first-order transition in the magnetization and possible related essential singularities. Thus, by analogy, the possibility of a magnetization jump is raised for the McCoy-Wu transition on a square lattice. The results for d1=1 are relevant to random quantum systems.