It seems odd that they’re constantly sharp. Two ideas off the top of my head…
1 - is it possible that the temperature in the room is changing drastically? Most instruments are subject to tuning issues as the strings change temperature. Banjos are extra sensitive because the head also tends to move, too.
2 - did you check your intonation? Maybe the strings are in tune when played open, but get sharp as you fret them?
https://banjobenclark.com/lessons/banjo-setup-with-steve-huber-bridge-placement-banjo?from_track=beginner-banjo