Ok, Oil cooler run in series with EGR Cooler. The oil cooler is water to oil, and uses coolant to remove heat from oil. This is transfered out to radiator by coolant. If oil cooler gets plugged, the EGR cooler(which cools the exhaust gas that is recirculated in your engine for emissions purposes), will overheat, thereby causing pressure spikes in the cooling system, thus causing coolant loss out of the overflow bottle.
Also, International uses "stretch" bolts to secure the heads to the block on these engines. They supposedly "improve clamping uniformity" during assembly, but in reality, all they do is end up "stretching" more, and over time and under heavy load, they literally allow the heads to "lift" off of the block during heavy loads. This allows combustion gas to enter your cooling system, also causing pressure spikes, air bubbles, loss of coolant, and overheating during loading. Pretty common problem. So common in fact, it one of the main things to repair to make a 6.0 reliable. The EGR heat issues just make the head problem worse.
So to fix your issue. you should do the following. Even if the gasket are not blown now, they eventually will, and can leave you stranded.
1. Replace head gaskets. Use ford oem parts, except for the head bolts. Replace these with ARP head stud kit.
2. Delete EGR system completely. There are vendors here that sell complete EGR Delete kits.
3. Install Coolant Filter system. This will prevent any gunk from getting into your oil cooler. Also when head gaskets are done, is good time to replace this.
4. (optional, but recommended) Reprogram computer with SCT tuning software. Eric at Innovative Diesel has one of the best reputations around. This will vastly improve the performance of your engine, reduce EGT during towing, and improve mpg.
This does not come cheap, but in the end, you'll have a completely different truck, and reliable to boot.