Nth for saga? Nth to see there so not worth? For hakata to yufuin, you mean taking JR or the scenic train?
The activity I did in Saga was Shopping at Tosu Premium Outlet.
I would suggest to skip Yufuin No Mori sightseeing trains (pretty hard to be available due to popular demand even on weekdays) and get the station staff recommendations and assistance to book available Sonic train & onward connections between
Oita & Yufuin eg. Yufu 3 / Yufu 5.