Pics are always good in situations like this, but you are very likely looking at the after effect of a ruptured oil cooler.
It isn't surprising that you don't see anything in your oil system. Your low pressure oil system is about 70psi and the coolant system is sub-16. That determines the path. You may notice a drop in oil level, though.
It's a kinda tricky spot. There could be other things going on, so it would be nice to know a bit more about the operating values of the engine. ...but it isn't so good to run it now.
The other tricky spot is this... The coolant system is now contaminated. All of the oil-laced coolant will need to be flushed from the system and - more than likely - a number of hoses replaced. Oil causes the rubber hoses to break down. You will notice gooey inside walls. Until the ruptured cooler is out, though, you are just going to be pushing more oil through.
My suggestions...
- With proper monitoring of coolant and oil temps, it is possible that you would've seen this coming. If you are not monitoring, start reading up on that.
- You might was well start stocking up on flushing/cooling system supplies. Coolant hoses, 30+ gallons of distilled water, CAT EC1-rated ELC, Cummins Restore and Restore+, a new degas cap and maybe a bottle, thermostat... Winter is coming.
- If you still have an EGR cooler, it is time to decide if you are going to keep it or ditch it (based on where you live and emissions). If you are going to keep, buy an upgraded cooler. If you are going to ditch, buy a delete.
...also - like the EGR question above - it is time to start thinking about what you are going to do and how you are going to do it. Given your catch-22 situation, this would be a great time to switch to either an air-to-oil cooler (BPD) or an oil cooler relocation (BPD, IPR).
Bullet Proof Diesel Heavy Duty Engine Oil Cooler
Coolant-to-Liquid Oil Coolers
IPR External Oil Cooler Kit for Ford 6.0 Powerstroke
That way you could clean things up as best you can, get the cooler out of there, and then go back to cleaning things up the rest of the way.
It would be good to know more about what you are comfortable with technically/mechanically. Sadly, you now have a problem that you will need to either wrench your way out of, buy your way out of, or sell your way out of.
If you are going to either wrench or buy, the guys here are the right ones of which to ask questions. They sure helped me. If you can tell a bit more about your truck, it will get things going.
Good luck!
Edited to add...
This writeup on flushing may help -
http://www.ficmrepair.com/pdf/Powerstroke_Cooling_System_Flush_v1-0.pdf
Here is info on Restore/Restore+ -
https://www.cumminsfiltration.com/pdfs/product_lit/emea_brochures/LI33024-GB.pdf
People also use Dawn or Cascade dishwasher soap to cut the oil. Restore is better, but not cheaper.