I use hot water on the opposite side if I can. If not, I use a low temp heat gun or hair dryer from a distance. It softens the adhesive.
The other trick is to pull slowly. The adhesive they usually use on labels is like toffee. It will break or become detached if pulled too fast. If you pull slowly and pace the pull rate to the relaxation rate of the adhesive, they will "usually" come off clean. Low heat and Patience is the order of the day.
If all else fails, Goo Gone won't hurt the plastic.
Goo Gone
Don't make the mistake of removing the small orifice from the bottle. It's perfectly designed to control the application rate.
Sometimes the goo gone will leave a bit of residue too from the dissolved adhesive. A little alcohol solves that problem.
I like the flambeau boxes too.
Dewalt has started making small stackable organizers now too. But they are also expensive.
The dollar store often has organizer trays. Ya, the trays and bins and cabinets all look like a broncs zoo, but...... they do work!