Originally Posted by bhomburg

Do not be patient. With pets - given the limited number of them accepted on flights - it is always advisable to book as early as possible. I`d strongly suggest you book ASAP to ensure you'll be able to travel on the dates you want.
There' s one flight a day from Seattle to Seoul: Flight number 271, operated by an Airbus A350, departing SEA at 2pm, arriving in ICN at 5:45pm. This flight remains the same all the time.
And Asiana has two flights a day from Seoul to Da Nang: one in the morning (Flight number 757), departing at 11:10 am, long before your connection from the US arrives; and one in the afternoon, (flight number 755) departing at 6:50pm, arriving at 9:35 pm. You need to book the afternoon flight (755) which will make for an easy transfer with a layover of an hour - long enough to give doggie a short break from the carrier.
Booking the morning flight the next day is definitely not recommended. This would make for a decidedly long overnight layover as you have noticed. You definitely don't want that.
There's nothing crisis-related about those flights, they've always been there.
Full service airlines like Asiana will not typically cancel flights with low bookings, no.
And Asiana doesn't look like they're poised to go out of business, either (a real probability these days for almost all airlines). They have secured billion-dollar funding to get them through the covid crisis.

If, and that's a big if, anything happens to your connection - say, the flight from Seattle is late and you can't make it - Asiana will rebook you on the next available flight, which will be the norning one the next day, and take care of you. The'yll arrange accommodation for you and your dog, don't worry.