Here is a way to produce examples with arbitrarily large densities.

Let $\mathbb{P}$ be any c.c.c. forcing that preserves CH, such as
the forcing to add a Cohen real, and let $\mathbb{Q}=\kappa^\ast$
be the decreasing linear order of length $\kappa$, a large regular
cardinal. Consider the product partial order
$\mathbb{P}\times\mathbb{Q}$ as a notion of forcing. This is still
c.c.c., since all conditions in the second coordinate are
compatible as it is linear. But every dense set must also be dense
in the second coordinate and therefore have size at least $\kappa$.
Since the second coordinate is trivial as a forcing notion,
the product $\mathbb{P}\times\mathbb{Q}$ is forcing equivalent to
$\mathbb{P}$, which preserves CH, and so
$\mathbb{P}\times\mathbb{Q}$ has all your desired properties.

This kind of example, however, may lead you to modify the question, since it is not separative.