When I came to have surgery for the first time, I carried all of my cash in a belt bag which I kept on me under my clothes at all times during the flight. I am Canadian, I'm not entirely sure how different the procedure is for Americans, but I had to declare if I was bringing more than 10 000$ Cad out of the country, which I did. I declared the amount at customs, didn't take long they only had me fill in a form. When I was entering the plane there were custom agents which were stopping random people and asking their names, I had a feeling they were trying to find me to question me into more details about the money, but I'm not sure. Anyways, I couldn't be bothered, and I managed to enter the plane without them paying much attention to me.
When I arrived in Korea, I did not declare any money because I had a bit less than 10 000 USD, so I didn't need to. Anyways, just make sure you don't lose the cash and you tell customs you're bringing cash with you and you'll be fine