In order to express not desiring or being able to ingest more food, the German word for “full” is satt. This word is related to the English “sated” and thus also to “insatiable” and “satisfaction”.
Evidently it is also related to the English “sad”, in the sense of “heavy” or “weary”.
The word satt can also be used when expressing being “fed up” with a situation.
Be sure NOT to describe yourself as voll (Ich bin so voll. – I am so full.) in German when refusing more nourishment because this usually expresses being very drunk.
Ich bin viel zu satt um das zu essen. I am much too full to eat that.
Ich bin so satt, ich mag kein Blatt. I am so full, I can’t eat another leaf. (well-known rhyming quotation from a goat in the German fairy tale ‘The Wishing-Table, The Gold-Ass, and The Cudgel in the Sack’)
Er ist total voll. He is totally drunk.
Ich habe dich total satt. I am totally sick of you.
Ich habe dein Theater satt. I am fed up with your drama.