No, because reals not in the ground model can be different in definable (with parameters from the ground model) ways.  For example, $\dot x$ might be (forced by all conditions to be) a Cohen-generic subset of $\omega$, while $\dot y$ is the intersection of $\dot x$ with the set of even numbers.  Then $\dot y$ is disjoint from an infinite ground-model set of natural numbers but $\dot x$ is not.