How would you handle drawer counts and cash drops at your job?

I've been in the hotel business for 25 years. I just started at a small hotel as a manager. Neither one of my employees have ever worked in a hotel before this one and they are having a hard time understanding how things are done the correct way because the previous manager had them do things all backwards in my opinion. Typically, every shift counts the drawer at the beginning of their shift. Then if you take cash you drop it in the safe yourself. At this place only the day shift counts the drawer and if anyone has a cash drop, they leave it up to the next day shift to do it. And they don't even put it in their own envelope they just add it to one envelope so there's a bunch of money in one envelope for the month. I've never worked at a hotel that did it this way.