I understand the frustration, that being said, it's not that hard to correct. First, contrary to previous editions, in 5e, unless I'm mistaken, once applied, the poison stays on the blade for one minute, which is way longer than most fights, so there should not be a reason to reapply it.
As for applying it, for the sake of fairness, I would require NPCs like drows and assassins to use bonus action to apply it, with the assumption that they did it before combat in most cases. After that, find additional doses is another matter. Poison is not that balanced in the game, never has been whatever the edition, sop allowing too many doses of very potent poison for PCs is unbalancing the game, and I'm not sure that there is much that can be done about this.
Some individual poisons differ, but the general rule for injury poisons is that they stay potent until delivered through a wound or washed off. However, in general they only work once, and then have to be reapplied, which is why the action type to reapply it matters. My Battlemaster/Thief could use a bonus action to apply poison, which I'd get on my first successful attack. Poison-using NPCs, however, get the bonus poison damage on all of their attacks without needing to reapply it, even when they get multiple attacks per round. So simply saying that the NPC applied it before battle doesn't work as an explanation after the first hit.