Hi @Daniel Twigden ,
To answer you initial question, app service slots do not share storage. A slot can be considered as a separate VM instance within your app service plan. A clear indicator of this is the fact you can't use slots for Shared VM SKUs.
You can share app configurations between slots or make an app setting be dedicated to a particular slot. You achieve this by selecting the 'Slot Setting' in Application Settings under the Configuration blade. Files on the other hand aren't shared as the instance is essentially a separate VM with separate storage.
This doc further explains what happens when you perform a swap.