zumi

A noise that became a verb, which is the trick Esperanto is best at. The dictionary has a part-of-speech tag for words like this, sonimito, a sound-imitation. Zum is the noise. Add -i and you have the verb the moment you need it; add -o and zumo is the buzz itself. The interjections and onomatopoeia article sets out the whole family: boji for a dog, miaŭi for a cat, plaŭdi for water, tiktaki for a clock. English has to wait for a dictionary to bless to meow; Esperanto derives it from the noise on the spot.

It covers both English words. A bee zumas and so does a fridge, and so do you when you sing with your mouth shut. English splits that into buzz and hum and then needs drone and whirr for the rest; zumi takes the lot, because what they have in common is a low continuous sound with no edges in it.

The bee is at 1,808. Abelo is the animal you will use this verb about most, and abelzumado is the sound of a hive, built from abelo plus zum plus the continuous-action -ad-. That word is not in any dictionary you own and every Esperanto speaker will understand it, which is roughly the point of the language.

Useful derivative. Zumilo is a buzzer, the thing on a door, from zum plus the tool suffix -il-.