The Copyright Office doesn't verify whether publication has occurred; they take your word that the information you've provided is accurate. That's because you've promised (Section 8, Certification) that "the work identified in this application and that the statements made by me in this application are correct to the best of my knowledge." You can publish your work during the period that the application is being processed. However, the benefits of registration would not apply to changes made in the published version.
Should you wait? As a general rule, if you know that you will soon be publishing your work, it's better to wait until you've published before registering it. That's because the version that's published is typically the "final" version, that is the one that is unlikely to be modified. Registration within three months of publication guarantees certain rights. As always, we remind you that you have copyright in the work without registration. Don't operate under the mistaken impression that it's a race to the Copyright Office (or that you should mail yourself a copy) to claim priority over other similar works that are registered after that date.