You will survive this.
And you can still get excellent training in psychiatry.
There are many good, even great, residencies that have a rising reputation - but students don't tend to send them applications because it takes decades for a residency to that kind of reputation.
Look for a residency that:
A) Is not in one of the 6-8 great cities of America. There are residencies with excellent training, but they are in cities that don't automatically attract everyone.
B) Recently hired several big name faculty away from other institutions. They are likely trying to revamp their dept.
C) Is near one the very best places to live in the US. Many good faculty (and adjunct faculty) will have moved to that area but the dept at the big premier university has no openings, and some of the best teachers are not researchers - so cannot get a full-time position at the university dept.
For instance, Pacific has a good residency in San Francisco, but it's not at the big university. Such residencies have a less than stellar reputation, but often have excellent overall training.
Honestly, I would give this advice to nearly anyone. It's my belief that the training at some of the "top" 15-20 residencies
1) suffer from the assumption that they are the best - so do not try as hard to keep improving
2) assume that any problems with the residency are due to the residents and not the program
3) abuse residents more and use them more as slave labor than programs that have to attract applications
4) attract some of the biggest names, but don't expect them to teach (or sometimes even see) residents. Top publishers rarely work in the wards or clinics - so having a psychiatrist (whose name is known to every psychiatry resident in the country) in the dept may mean nothing to the residency, except that the dept has expended much of its resources on an asset to which residents will not have access.