During a loan review we have determined that in two different loan transactions, the appraisal fees were disclosed $100 more than they should have been (new loan processor).

The FAQs speak more on how to disclose a tolerance violation cure. This was not a tolerance violation. So my question is, do we just amend/correct the HUD to disclose the actual cost of the appraisal and send a nice apologetic cover letter with corrected loan docs and a cashier’s check? Does this correction have to be disclosed in some special way like is done for tolerance violation cures?

I'm assuming you're talking about overdisclosing on the HUD.
The only 'loan docs' you'd correct would be the HUD.
Just correct the figure and send them a check for the overpayment along with a letter of explanation that it was a clerical error.

Thanks

Are the amended HUD and check required? Or would it suffice to credit the loan when there is no tolerance issue?

No, you cannot just credit the loan. How is the borrower going to know this was done or why? Further, this is money you should never have collected. It's up to the borrower to decide what they want to do with their money.
You must correct the HUD to properly reflect the charges paid by the borrower and deliver a copy to them, along with the money they overpaid.

So is a corrected HUD always required? If you have a loan that is closed with the understanding of it being sold on the secondary market, fees are collected they charge, then they reject it for one reason or another. It is then booked in house. Those fees that were collected as required by the secondary market are now not required, but were collected. Can we send them a "Hello" letter basically saying we are going to keep your mortgage here now, and these fees are not required, but we collected, and send them a refund check? It was a purchase transaction and closed by an attorney.

I'm not sure what specific fees you might be referring too. What third party fees would now not be paid??