All engines create pulsations in their crankshaft/flywheel output to the transmission, if coupled directly to the trans, a lot of gear noise is created.
The Dual Mass Flywheel (DMF) is designed to minimize gear noise among the other normal clutch things it does, but to accomplish that is has a LOT OF PARTS in it, typical for a Ford 7.3L about 120 individual components down to the rivet. I have taken several apart.
A correctly functioning DMF and a non-worn out transmission are the quietest combination that your truck could ever have.
This is accomplished by two seperate functions inside the DMF, first a torsion damper comprised of a series of coil springs that ALL of the engine power is transfered to and by its ability to be compressed and rebound, it smooths out the power pulses significantly. The second element is that the secondary (friction side) of the DMF is considered to attach its weight directly to the input shaft this adds flywheel effect to the input shaft and it also minimizes the pulstations that trigger the gear noise that you do not want to hear.
The DMF needed a public relations campain manager shortly after it hit the 7.3L engines with all of the tales of woe and doom, many of which were related to miss-understanding of its functions and CORRECT service procedures, for example you NEVER want to dip or spray a lot of solvent or cleaning solution on the Ford DMF due to the internal friction torque limiting clutch inside the DMF.
Several aftemarket clutch companies responded to the DMF shortcomings and offered a Solid Flywheel (SF) conversion. The SF system cannot exactly match the filtering capabilities of the DMF so under some loads/RPM/gear conditions you hear more noise than normal from the DMF. The trade off is cost vs less cost, 120 pieces vs a solid cast iron SF and a simpler disc that does a darn good job for most all customers. The internal clutch of the DMF also limited the power that can be transferred to the trans which has saved many trnasmissions, but ticked off a lot of the power hungry Powerstroke owners, this also drove the SF segment.
Fact: The company that built your Ford DMF made the last production run, no more will be built.
A DMF cannot make up for the noise a worn transmission creates, it will not make gear teeth grow back or fix worn bearings.
The tech support for the clutch company that supplied the parts should have been contacted PRIOR to the teardown of the transmission.
If your supplier has multiple clutch systems, was the correct system chosen for a replacement clutch? Some are replacement SF systems and sometimes a more agressive and stronger system is chosen which may have an even cruder simple torsion damper that is more noise prone, but strong as you will ever need.
I do not recommend seeking a solution to your problem by adding more or a different type of oil, just use the type and qty. that Ford spec'd out.
Also, make sure that all boots at the shifter are in place and not popped out of place.
Your question is not a new one to the clutch industry, and FWIW, I do not work for the compan that made the DMF.
2005 Dodge Ram 2500 5.9L 6 Speed
16" Michelins, Hi Spec Wheels, Max Brake, Carslile Actuator, Equal-i-zer, Dill TPMS. Campfire cook.