My inquiry stems from a recent thought that "decade" and "decadence" might be related. Does anyone know if the two words share the same or similar origin?
Not really. Decade is from Latin decas, decadis, from Greek dekas and is related to decem 'ten', while decadence is from Latin decadentia < de- 'from, away' + cado, cadere 'to fall' (< PIE *kad- 'fall') whence case, accident, etc.