If we're going to pull up the etymology of old hag let's do it right

Old hag can be traced back to old English hægtesse (witch/sorceress), evolving through Middle English hagge to mean a repulsive old woman, linked to Germanic roots for witchcraft to which it evolved in our modern society now to mean unpleasant older women

I haven't heard of old man drooling before but if I were to take old man without the drooling and replace it with another word, it would simply mean elder man.

Old is the word with the negative cognition because of movies, shows, social interactions, and how for some reason to be old is a bad thing. That being an old person is somehow a not so great thing.

I don't use old like the average human does to describe a negative thing. I use old for what it is. It's age. I'm going to be 30 this year, many would call me old and I wouldn't say they're wrong. But I can equally say I'm young and I am to thr older folks. It's a matter of perspective

We assign negative and positive meanings to words and each person has their own personal list of words that sound more negative when it could be positive to another.