In this paper, we consider the problem of variable-length coding over the class of memoryless binary asymmetric channels (BACs) with noiseless feedback, including the binary symmetric channel (BSC) as a special case. In 2012, Naghshvar et al. introduced an encoding scheme, which we refer to as the small-enough-difference (SED) encoder, which asymptotically achieves both capacity and Burnashev's optimal error exponent for symmetric binary-input channels. Building on the work of Naghshvar et al., this paper extends the SED encoding scheme to the class of BACs and develops a non-asymptotic upper bound on the average blocklength that is shown to achieve both capacity and the optimal error exponent. For the specific case of the BSC, we develop an additional non-asymptotic bound using a two-phase analysis that leverages both a submartingale synthesis and a Markov chain time of first passage analysis. For the BSC with capacity $1/2$, both new achievability bounds exceed the achievability bound of Polyanskiy et al. for a system limited to stop-feedback codes.