We revisit the MaxSAT problem in the data stream model. In this problem, the stream consists of $m$ clauses that are disjunctions of literals drawn from $n$ Boolean variables. The objective is to find an assignment to the variables that maximizes the number of satisfied clauses. Chou et al. (FOCS 2020) showed that $Ω(\sqrt{n})$ space is necessary to yield a $\sqrt{2}/2+ε$ approximation of the optimum value; they also presented an algorithm that yields a $\sqrt{2}/2-ε$ approximation of the optimum value using $O(\log n/ε^2)$ space. In this paper, we focus not only on approximating the optimum value, but also on obtaining the corresponding Boolean assignment using sublinear $o(mn)$ space. We present randomized single-pass algorithms that w.h.p. yield: 1) A $1-ε$ approximation using $\tilde{O}(n/ε^3)$ space and exponential post-processing time and 2) A $3/4-ε$ approximation using $\tilde{O}(n/ε)$ space and polynomial post-processing time. Our ideas also extend to dynamic streams. On the other hand, we show that the streaming kSAT problem that asks to decide whether one can satisfy all size-$k$ input clauses must use $Ω(n^k)$ space. We also consider other related problems in this setting.