It seems you assume $X$ is independent of $W_i$, and $W_i$ are jointly normal. Following this fact, $Z_i$ become jointly normal, and you can use the [conditional distribution][1] formula to find the joint PDF of $Z_i$. A similar approach for finding the joint PDF would be directly calculating the mean and covariance vector for $[Z_1, Z_2]$ ($Z$ is jointly normal because it's a linear transform of the random vector $[X,W_1,W_2]$). However, multivariate normal CDF has [no closed form][2].


  [1]: https://en.wikipedia.org/wiki/Multivariate_normal_distribution#/Conditional_distributions
  [2]: https://en.wikipedia.org/wiki/Multivariate_normal_distribution#/Cumulative_distribution_function