During the peak travel time, even Shinkansen trains with each train having 1323 seats can sell out in a minute or two.
Yes, Yufuin is a very popular tourist destination, and Yufuin-no-mori express train is a very popular train for tourists heading to Yofuin. Besides, you are looking at travel on Oct. 27 and 28 (weekend travel) where the peak foliage season of the Yufuin area is from the end of October into November. I am not completely surprised to hear Yufuin-no-mori express train sold out immediately after the start of the ticket sales. You are trying to get a seat on the Yufuin-no-mori express train during one of the peak tourist seasons of the Yufuin area for weekend travel.