- leaving on April 1 is certainly not last minute. Once we have managed to book a leisure trip to Tokyo from East Coast in < 48 hr before departure (family of 3)...
Regarding your trouble with booking these tickets on aa.com, I think the problem is an ongoing update to AA website/booking engine that does not work as intended. I suggest you try Japanese AA website (available in English) - that site is showing some excellent prices for April 1 departure (e.g., 77,980 JPY for April 1-9). That site would book through Amedeus whereas US version would be Sabre. You can also try OTAs to book your flights. Good luck!