If the transformer tank cannot take significant vacuum, you could still rig up some closed ductwork that contains some household dehumidifiers. These are quite efficient at drying stuff. I have used them to dry up rooms that have moisture problems and even raw lumber. The dehumidifiers will provide a moderate amount of heat that will not cook the windings.
Since most modern transformer oils are of reduced flammability with a very high flash point and ignition point, you probably do not need explosionproof stuff. Check the material safety data sheet for your oil. Since you need to keep out oxygen for the transformer's sake, slight pressurization with dry nitrogen should take care of any explosion hazard, what is known as purged and pressurized protection against explosion hazard.
When you reach the limit of what dehumidifiers can do you can also pass the atmosphere through silica gel cat litter, or at least that is one way to buy lots of it.
You probably need to pass the bulk oil through a filter and drier. I would recommend draining the tank through a filter and drier and then using a fresh filter and drier for refilling. Just remember do withdraw and replace the oil slowly. Too much oil velocity will stir up sediment and also cause the oil to develope static electricity bubbles that can damage the transformer. Use grounded metal pipe to connect your filters and driers and the oil hose.