Hi @clarajones - It's difficult to tell from a screenshot, but if building new trade depots helps for a little, I would guess that your storage is full. So, the error might be saying "needs oil" but should really say something like "there's a problem with oil in this city".
It might take some investigation to see what the real issue is. Do you have your depots or ports set to sell the oil? If so, make sure your traffic flow is good. Selling a resource sends trucks in from "the region", and if they get stuck in traffic they won't be able to leave the city, or it will take a lot longer. I like to build my trade depots and ports close to an entrance, on a side street where the trucks can get in and out easily. Check your budget sheet to see if you're getting cash for your oil regularly.
Do you have your depots or ports set to "use locally"? This is where you want to be sure you have added delivery truck garages. Transportation is always an issue here, too, make sure the trucks can get from the depots to where the oil is being used. Then, make certain that you're not drilling more oil than you can use. You can always turn off a few pumps and let everything catch up so that you don't have the deliveries stalling all the time.
These are just a few ideas. I would encourage you to do the same sort of troubleshooting with your other cities. Look for issues in your supply line. You may have plenty of a resource, but it's not getting where it needs to go. Check your other maps - especially the trade maps, to see where your trucks are, and whatever industries you may be supplying with the resource.
Hope this helps! Happy Building!