In this paper, we design erasure-correcting codes for channels with burst and random erasures, when a strict decoding delay constraint is in place. We consider the sliding-window-based packet erasure model proposed by Badr et al., where any time-window of width $w$ contains either up to $a$ random erasures or an erasure burst of length at most $b$. One needs to recover any erased packet, where erasures are as per the channel model, with a strict decoding delay deadline of $τ$ time slots. Presently existing rate-optimal constructions in the literature require, in general, a field-size which grows exponential in $τ$, for a constant $\frac{a}τ$. In this work, we present a new rate-optimal code construction covering all channel and delay parameters, which requires an $O(τ^2)$ field-size. As a special case, when $(b-a)=1$, we have a field-size linear in $τ$. We also present three other constructions having linear field-size, under certain constraints on channel and decoding delay parameters. As a corollary, we obtain low field-size, rate-optimal convolutional codes for any given column distance and column span. Simulations indicate that the newly proposed streaming code constructions offer lower packet-loss probabilities compared to existing schemes, for selected instances of Gilbert-Elliott and Fritchman channels.