sorry and sorrow
are not related! Not etymologically related anyway. They were associated with each other phonologically and semantically in Middle English.
sorry and sore are from PIE *seh₂i– “suffering” (Old English sār “painful” and sārig “distressed, sad” (cognate with West Frisian searich “sore, spotty, scabby”)).
sorrow is from PIE *swergh– “worry, be sick” (Old English sorg “anxiety, sorrow”).
In a similar vein, my wife asked me this morning if the two senses of meal were related, and I explained that they are not: meal ‘ground seeds of a cereal grass or pulse’ is from PIE *mel(ə)- ‘crush, grind’ (and is cognate with German Mehl, Dutch meel, and Old Norse mjǫl and related to Latin molere ‘to grind’), while meal ‘the food served and eaten on a particular occasion’ is from PIE *mē– ‘measure’ (and is cognate with German Mal ‘time,’ Mahl ‘meal,’ Old Norse māl, and Gothic mēl ‘time, hour’ and more distantly with moon, month, and meter). (The semantic transition is from ‘measure’ to ‘marked out or appointed time’ to ‘time for eating’; there’s an obsolete phrase sorry meal meaning ‘an untoward, terrible, or unhappy occasion or occurrence.’)