I am working with a space with the following properties, and want to know if it is necessarily metrizable.
I ran this through pi-base and found no examples which were not metrizable. For what it's worth, if the first property is replaces with mere $\sigma$-compactness, there are non-metrizable examples.
The space $X=\mathbb Q\times\{0,1\}^\mathfrak c$ is a counterexample.
$X$ is the union of the compact nowhere dense sets $\{r\}\times\{0,1\}^\mathfrak c,\ r\in\mathbb Q.$
$X$ is Lindelöf because it's $\sigma$-compact.
$X$ is $\text T_3$ because it's a product of $\text T_3$ spaces.
$X$ is $\text T_4$ (and paracompact) because it's a Lindelöf $\text T_3$ space.
$X$ is separable because it's the product of $\mathfrak c$ separable spaces.
$X$ is not metrizable because $\{0,1\}^\mathfrak c$ is not first-countable.
More generally, let $Y$ be any $\text T_3$ space which is $\sigma$-compact, separable, and non-metrizable. (You indicated on the last line of your question that you have examples of such spaces. For example, $Y$ could be a countable $\text T_3$ space which is not first-countable.) Then $\mathbb Q\times Y$ is a non-metrizable space with all the properties you want.
