Oil in Propane

Not odd at all. Propane is "refined" from oil and sometimes a bit of oil remains in the finished product and is transferred to the customer's tank.

As for the advice to go to a Big Propane place like Amerigas et-al.. (I like Parker's in Flint, MI or Carmichael's in Tekonsha MI) The wholesale places (Quick Stop in Quartzsite AZ is good,, you will find a lot of smaller places that are basically outlets and do not do repair work You need a tank cert place to drain, purge and refill the tank... Sounds worse than it is.

That one in Tekonsha, MI (Home of the Prodgy Brake Controllers) can be hard to find for most folks (For me it's dang near "With eyes closed" 60 years ago it was a dairy farm where (This is a joke folks) I shoveled political promises or something like them.

Odd one...

What kind of oil exactly did it seem like? motor oil? cooking oil? or just a black ooze?

There is no reason oil should be in a propane tank. If there is oil then I can't see how it would matter. The propane comes out of the tank as a gas, and if there is oil in that tank it should not vaporize and come out with the propane gas.
Oil in the copper tubing on the camper carrying the propane should not matter either. It should settle out or just stick to the walls and never make it to the appliances.
If oil somehow made it to the burners then what would it do? burn?

I suspect your oil could be a deteriorting rubber hose.

As for the question, how to flush a tank... It will be cheaper to just get a new tank. I assume you have a 30 lb tank, not an exchangeable 20 lb bbq tank.
You cannot flush it. A qualified propane tank business will be able to remove the nozzle and then have full access to the inside of the tank, or maybe they have an easier method.

Flushing your plumbing system? No need that I am aware of unless it is full oil quarts of oil, and if that is the case you have some serious problems that cannot be solved with advice.

Propane dealer, like Suburban or Amerigas.