Differential privacy for the 2020 census required an estimated 90 terabytes of randomness [GL20], an amount which may be prohibitively expensive or entirely infeasible to generate. Motivated by these practical concerns, [CSV25] initiated the study of the randomness complexity of differential privacy, and in particular, the randomness complexity of $d$ counting queries. This is the task of outputting the number of entries in a dataset that satisfy predicates $\mathcal{P}_1, \dots, \mathcal{P}_d$ respectively. They showed the rather surprising fact that though any reasonably accurate, $\varepsilon$-differentially private mechanism for one counting query requires $1-O(\varepsilon)$ bits of randomness in expectation, there exists a fairly accurate mechanism for $d$ counting queries which requires only $O(\log d)$ bits of randomness in expectation. The mechanism of [CSV25] is inefficient (not polynomial time) and relies on a combinatorial object known as rounding schemes. Here, we give a polynomial time mechanism which achieves nearly the same randomness complexity versus accuracy tradeoff as that of [CSV25]. Our construction is based on the following simple observation: after a randomized shift of the answer to each counting query, the answer to many counting queries remains the same regardless of whether we add noise to that coordinate or not. This allows us to forgo the step of adding noise to the result of many counting queries. Our mechanism does not make use of rounding schemes. Therefore, it provides a different -- and, in our opinion, clearer -- insight into the origins of the randomness savings that can be obtained by batching $d$ counting queries.