I'm confused: are you trying to cover the engraving or not?

My experience thus far with Duracoat shows that it does not tend to be a good filler - I've done a few magazines, and the date stamps and floor plates are still perfectly legible. I suspect you'd find the same results with engraving on a 37.

The regular duracoat coating will not fill in the engraving. I've got a Remington 1100, with similar engraving to the Ithaca 37, that has 5 coats of duracoat on it, and the engraving is still clearly visible. Maybe the duracoat "filler" will do it? http://www.lauerweaponry.com/index.cfm?fuseaction=category.display&category_id=915

JB Weld works too.

I have an old Erma-Wehrke .25acp auto pistol that had some major pitting on one side of the slide, when I inherited it. I hand sanded it as smooth as possible, then filled in the pitting "craters" with JB Weld (they were much deeper than the engraving on a 37... I've got a 20ga Featherlight myself, so I know what it looks like). After it sat in the sun a few days and the JB was good and hard, I sanded it smooth again, then Duracoated it in black. Looks like new.

to fill the engraving with duracoat you would need to coat it, let it cure, use a flat block and sand it down, repeat until the engraving is filled in.

Here is the completed picture. Once I got the thing sandblasted the engraving really didn't bother me too much so I just ended up leaving it alone. Cut the barrel to 18.5 inches, just to be on the safe side, and finished it off with Duracoat Extreme Tactical Grey. This was my first shotgun and had killed many a rabbit but it was just sitting there not getting used and looking pathetic. Now it has been repurposed and looks fantastic. I also never knew about slam firing until I joined this website so being that it is a 1971 model I can't wait to try that out!