We study differentially private approximation algorithms for positive linear programs (LPs with nonnegative coefficients and variables), focusing on the fundamental families of packing, covering, and mixed packing-covering formulations. We focus on the high-sensitivity, constraint-private regime of Hsu-Roth-Roughgarden-Ullman (ICALP 2014), where neighboring instances may differ by an arbitrary single constraint, so one cannot hope to approximately satisfy every constraint under privacy. We give private solvers that return approximate solutions while violating only a controlled number of constraints. Our algorithms improve the prior instance-dependent guarantees, and also yield new data-independent bounds that depend only on the dimension. Our techniques involve a dense multiplicative weights update method developed from a regularized dual viewpoint, which we analyze in a way that exploits structure specific to positive LPs.