Tom,
I, like you, used to hunt with an Ithaca 37, deer & small game, it was my first gun, and I did a lot of hunting with it as a kid, so a pump just seems to come naturally to me anymore. My son shoots that Ithaca 37 today, my first gun, handed down to him.
I now shoot an 870 12 gauge pump for deer, and a 760 in 30-06. The 870 has a 1.5 x 4.5 Bushnell Banner scope mounted to the receiver via a 760 weaver base(receiver drilled & tapped by myself), despite the fact that everyone said I was crazy to drill & tap a brand new gun in 1980, LOL. The 760 has a 1.5 x 5 Leupold vari-x III on it, LOW rings on both guns. I have the barrel band style of Swivel studs on the 760 so I could make the swivel studs the same distance apart on both guns, and use an Uncle Mikes quick adjust sling, which I put an additional piece on to extend it, so I can shoot military style offhand with heavy clothes on for rock solid stability.
I never had any trouble adjusting from one gun to the other, and tend to use them interchangeably, as I use the shotgun in New Jersey, and the rifle in New York(shotgun & muzzleloader only in New Jersey for deer). I will also use the shotgun in New York if conditions warrant it.
As far as adjusting from the shotgun to the rifle, or vice versa, I have never had any problems, and personally view a shotgun shooting slugs as more of a limited range rifle than a shotgun anyway. Just my personal thought and opinions.