Oil cooler.
I did not have one...but the manual for my '96 shows it.
The oil cooler, if it is there is a small unit that fits onto the oil filter mount...and the oil filter screws into the cooler.
In other words.....it is a plate that is about 1" that is sandwiched between the oil filter and the place the rest of us put the oil filter onto.
The ONLY way that I would be able to know if I had the oil cooler on the engine is to look for the pipes on it....which coolant hoses connect to.
If you have coolant hoses going to pipes that seem to go into the base plate that the oil filter screws onto......then you have the factory engine oil cooler.
If you don't have the coolant lines there, you do not have the vactory engine oil cooler.
I am also thinking that you do not have a bad head gasket....
I would lean toward the front cover.....also known as timing cover.
Sadly, it is a lot of work to replace that gasket......cheap part....and the new ones are improved, so that they will not fail (if installed correctly) but it is a MAJOR repair.
I had the front cover gasket start to leak VERY slightly......coolant leaking out....a FORD dealership told me not to worry about it unless it got worse....that it was a $1200 repair.
At the time.....the repair process started with "remove engine from vehicle".
They have modified that now......but you still have a LOT of work.
What I did was put the original pelleted BAR's Stop Leak into the radiator.
That sealed it up for MANY MANY miles and years.
The Bar's did NOT solve a leaking lower intake manifold gasket (although it did slow it down....again.....it was a very slight leak also....but coolant into the oil).