My charcoal supply went to 0, despite green level lumber supplies in all 4 of my charcoal factories...

When you view the charcoal status you will see a production rate and a consumption rate.. The more buildings you have and the higher you upgrade them the greater your consumption rate will be.

If it is greater than the production rate then your total charcoal will still go down even if you have lumber in the charcoal factories.

You need to upgrade the factories and also the Charcoal workshop building which produces charcoal without needing lumber. Upgrading the factories will make them work more efficiently and let you produce charcoal faster, this will improve your production rate.

Disconnect all buildings that do not need to be connected to the main castle. Disconnect the roads. Only connect your worker huts, production buildings, storages, troop training (whichever you want to produce), blacksmith, dragon altar. That's it. Disconnect all the other buildings (Including Tavern) by cutting off the roads leading to those buildings. If you need to upgrade one of those buildings, just connect with a road, start upgrade, then disconnect.

By doping this, you will effectively reduce the charcoal consumption by at least 50%. I have seen some people disconnect Farms and Ale houses also to further reduce charcoal consumption.

One thing to keep in mind - The higher level a building, the higher the charcoal consumption. So, if you do not really need a building in high level, do not upgrade it. Make sure to keep the Tavern disconnected at all times because this is the worst charcoal consumer. Definitely max the Charcoal Workshop. That will increase charcoal production. Definitely upgrade charcoal factories, which will improve charcoal production rate and efficiency. Dragonite workshops and Crystal workshops do not consume charcoal. So, no need to disconnected those.

Goodluck