Agree with Frederica, this has nothing to do with ACGME. Let's assume you (or admin) meant that contacting programs would be an NRMP match violation.
That's ridiculous. There's no violation, not even close. Your dean could call each PD individually, or send cookies, or anything, and it's totally fine.
The fact that their first answer to you wasn't "Holy crap, this is our fault, let's try to fix it ASAP" is bad news. Because it (mostly) is their fault, assuming all of the letters were correctly attached to cover letters etc.
Uploading the correct letters with a cover letter is a potential solution, but you're correct it is unlikely to have much positive impact. In fact, there's some chance that it might bring attention to the error that someone has (somehow) missed in reviewing your application. Even if they do upload new letters, the old ones remain visible and the new ones get addended to the bottom. So, overall, not very helpful.
As I mentioned already, there is no great solution to your problem. Applying to new programs with corrected letters might work, but it's very late. If you do this, create NEW slots in ERAS for the letters to be uploaded again -- as if they "fix" the current letters it will only append a copy of the new letter to the old letter.
Best option at this point:
Insist on meeting with the Dean of Students, perhaps with the Dean of the school personally.
1. Consider applying to new programs with fixed letters. Your school should pay ERAS fees for all new applications.
2. Have Dean contact these programs personally, explain the screw up, take responsibility for it, ask them to evaluate your application despite late submission.
3. Have Dean contact all of your 1st choice specialty programs and explain the situation.
Fork in the road: Will your school give you free tuition and COL expenses for another year?
Yes: Then cancel all of your second choice specialty interviews. Enter the match for your 1st choice specialty. If you match, you're done [although you could consider asking your school to reimburse you all ERAS/NRMP/Interview expenses for the catastrophe]. If not, then you complete a 5th year and match again next year with a new application, unlikely that anyone will remember you from this year.
No: You're in a tough spot. I would consider getting a lawyer -- the goal is not to go to court, but to force the above plan. Still stay in the match for your 1st choice specialty. You have to decide whether to rank the 2nd choice specialty, vs not matching and doing research and applying again, vs getting a prelim spot.