We consider the two scenarios of communicating a pair $S_{1},S_{2}$ of correlated sources over multiple access (MAC) and interference channels (IC) respectively. We undertake a Shannon theoretic study and focus on achievability, i.e., characterizing sufficient conditions. In the absence of a Gaćs-Körner-Witsenhausen common part, the current known single-letter (S-L) coding schemes are constrained to the S-L long Markov Chain (LMC) $X_{1}-S_{1}-S_{2}-X_{2}$. Taking the lead of Dueck's example [Dueck, Mar 1981], we recognize that the latter constraint is debilitating, leading to sub-optimality of S-L coding schemes. The goal of our work is to design a coding scheme wherein (i) the choice of channel input at time $t$ is based on multiple source symbols, and is yet ii) amenable to performance characterization via S-L expressions. In this article, we present the first part of our findings. We propose a new separation-based coding scheme based on a fixed block-length (B-L) codes that enables choice of $X_{jt}$ - the symbol input on the channel by encoder $j$ at time $t$ - to be based on a generic number $l$ of source symbols $S_{j}^{l}$, thus permitting correlation of the input symbols $X_{1},X_{2}$ through a multi-letter LMC $X_{1}-S_{1}^{l}-S_{2}^{l}-X_{2}$. By carefully stitching together S-L coding techniques we devise a multi-letter coding scheme. We characterize an inner bound to its performance via a S-L expression and prove that the derived inner bound is strictly larger than the current known largest inner bounds for both the MAC and IC problems based on S-L coding schemes. In the second part of our work, we propose to enlarge the inner bound derived in this article by incorporating the technique of inducing source correlation onto channel inputs [Cover, El Gamal and Salehi, Nov 1980].