When SCRABBLE for Android is idle - either in the games list or showing a game board - the CPU usage can get quite high. SCRABBLE can be the dominant user of mill time for extended periods, quickly flattening the battery. This has happened before and was fixed fairly promptly.
Whilst some usage is to be expected whilst SCRABBLE is doing 'nothing', this appears to be excessive. Android highlights the app in red some of the time. Usage is typically several times higher than other apps in memory at the time.
I have to ensure that I quit the SCRABBLE app on Android to stop the drain. It makes me reluctant to use the game.
Thanks for this detailed report @alanjenney. May I ask the model of your device as well as the version and number of the OS?