You are more then welcome, my passion is these old trucks and giving back for what I have received.
10W-30 diesel oil or 15W-40 will work. Many many many diesel owners run Rotella diesel oil, I don't put my money with shell as quickly as I put my money with Mobil so I run Delvac 15W-40 conventional in my diesels.
My wife and I own 3ea Ford Powerstroke diesels here, all 7.3 turbo diesels. They use 15 quarts of oil and it gets changed every 5000-7500 miles ($$$$). So instead of running full synthetic which would cost me about $125-160 per oil change I run conventional Mobil "delvac" 15-40 in our trucks which cost more like $45-60.
I have been very happy with the price and performance of the mobil oil, big shock, in my lifetime Mobil one has always been the best fluid.
We have a 01 Excursion 7.3, a 2000 E350 7.3 and my shop truck is a 97 F350 4 door 7.3 with intercooler and swamps injection. I love my turbo diesels!
So 10W30 or 15W-40 will work in your engine just fine.
You can run the full 5 quarts of diesel oil in your truck for a full oil change. However since you are doing head gaskets and fighting a carbon buildup issue, I would plan to run a full 5 quarts of diesel oil for a couple of weeks or about 1000 miles....then drain it and change the filter. Do it again... this time go the full 3000 miles and drain, change filter
Do it again! another 3000 mile full oil run using diesel oil
Diesel oil will not hurt your engine it just has WAY more carbon fighting detergents in it than gas oil does.
The reason I say switch to mobil one full synthetic after that is because the mobil is very good and removing carbon and keeping it gone.
I am sure other brands are too, just in my opinion mobil is the best and strongest.
You can switch to whatever oil you would like after that, hopefully the carbon is broken up and whatever is plugging your system is broken up to little bits and flushed out.
I run valvoline full synth in my personal fords simply because Mobil one is another $25 per change and I change my oil at 3000-5000 miles, the valvoline is a bit cheaper.
Motocraft semi synthetic is also a good oil at a good price, much of my family runs that.
While you are in there working on the engine you can fill the crankcase with ATF or diesel oil and I mean FILL the crankcase up to the top of the crank....let it sit in there until you are done with the head gasket job...then drain it out. This will give a nice head start to busting up the chunks of hard carbon that sticks to every passage
I never have been a big fan of "cleaning the crud out".
Those deposits stuck to the block etc. are enough to
score and kill bearings when flushed down.
Oil pick-up screens can't be cleaned with the oil pan on
if it has one....
Well of course not! and I agree
As Gary points out the best method to save your old 4.0 here would be to pull it, strip it down to the short block and remove as much carbon as you can. Clean the oil pan, pickup tube/screen. check the condition of the oil pump and gasket, have a look at the crank bearings, etc......
This is not the situation you are in with the engine in the truck and the heads off. I understand this as I have been there quite a few times myself.
I have opened enough 4.0 and 5.0 engines to know that carbon builds up when engines get hot and the oil is not changed frequently enough.
When we get a low oil pressure situation in a pushrod 4.0 or 5.0 it is ALMOST ALWAYS carbon buildup.
Yes the bits of carbon can get flushed down into the crank case and hurt your bottom end. That is why you want to change it at 1000 miles (or sooner) and soon there after whenever you add a detergent oil (or any carbon bustin detergents)
These engines have very very strong bottom ends and can run for a long time even after these "low oil pressure" situations IF you get the passages unplugged!
Right now with 6psi at warm idle you are already on borrowed time and there is already carbon buildup in there.
A good flush procedure can buy you another 100K miles+
I have seen it with my own eyes more then once, a engine that has 0 oil pressure at warm idle comeback to have almost full oil pressure after this treatment.
A few years ago we had just replaced the auto transmission in a 2000 mountaineer that had 278K on it, we put a used transmission in it.
The engine lost oil pressure soon after the trans was installed. So to save the truck my friend used the detergent oil process we are discussing here.....last I checked that mounty still driving around as a daily commuter