Yes, their office will send a Congressional inquiry to the DoS location involved. I work for a Federal agency and have to deal with receiving this type of inquiry all the time...sometimes it's crackpot grievances but not always and it can shake things loose. Have all of your facts and information, clearly set forth the timeline; be factual in describing your issue and the impact to you but don't be emotional or ranting, as that won't help.

Since you've already submitted a package, and the DoS location apparently doesn't deny they have it, it seems like this would be better than trying to start over, which could cause more confusion perhaps.

In the meantime, when you contact the location that's supposed to be handling your application, have you escalated the issue and specifically asked to speak with a supervisor or higher-level person? If not, do that as well. Again, be assertive but unemotional and polite, but insist on escalating the problem.