The oldest selected sense concerns fullness, not sorrow.
Old English sæd described being sated. Middle English sad had senses including serious or solemn, while sorrowful usage is recorded by around 1300. The exact route between these meanings is uncertain.
This samples attested senses, not a proven chain of causes. The later senses overlapped; dates are not boundaries between exclusive meanings.
Sources
- Merriam-Webster: sad
Word History and definitions support the ancestry and selected senses.
- Online Etymology Dictionary: sad
The entry gives sated, serious and sorrowful senses and explicitly discusses alternative explanations for their development.