I think you can "come close" to your objective, especially if hunting is not in the forefront of your intended uses. I'm. with those that say get your .44 "properly" fixed--or even get another (of some variety) if you're soured on that particular one. Mine's a pair of Rossi 92s. I'm the first to admit they're not the ones to optimally scope, but not why I have them to begin with. Nothing wrong with a Marlin either, properly set uo.
Still, the .44's not the reliable deer-getter beyond 150 or so, if that's an issue. If it is, the .30-30's a better choice--on up to 200-225 max, or if you're dabbling in those ranges and beyond, I'd echo comments on nixxing the HD role--as you do technically have it covered--and getting a .308 or 7mm-08. The latter is particularly tolerant of shorter barrels --as in 18-20" (18" Remington Model 7 comes to mind as one particularly nice application), and of course, being a descendant of the .308 is a short action.
But, as you say no dedicated hunting rifles, back to the .44, it's a fairly versatile combo with the lever. 44 specials or specialized SD rounds for close-in home defense duty, and as hot as you want for the short range hunting and "black bear emergency" duty. I always have my .44 lever and companion revolver handy in camp in such bear (and lion) country in Arizona.
Another choice - Ruger Mini 30. Though it's been called basically a semi-auto .30-30, I'd choose literally a .30-30 over it for strictly hunting duty--especially with its array of heavier bullet choices--but as a dual-duty gun the Mini 30 certainly would fill the bill. A lot has been said about inaccuracy in the Mini 30s in its first few decades of existence, but it's fine for minute-of-critter, and that aspect is supposedly improved in the newer models. Also, IIRC Ruger's also finally out with hi-cap mags for more home-defensive duty and range fun. You'll hear a lot (maybe here) about how the SKS or even AK is a much better buy in the same round. Maybe so, but especially the former has about as much charm as a brick, and re the latter the Mini 30s a bit "more PC" in appearance. Either way, over-penetration certainly would be an issue with the 7.62x39, more so than a relatively mildly loaded-but-effective and blunt-nosed .44. The .44 Special is one of the very best SD rounds out of a revolver, very similar ballistics to the vaunted .45ACP. So, you can imagine decent effectiveness of 8 or 10 such rounds out of a 16"-20" tube.
In sum, get/keep the .44. If you then get the hankering for more serious mid-to-long range deer or elk (etc) hunting, graduate up to an approporiate bolt-action (.308/.30-06 class again).
Later EDIT. I noted the .30-30 briefly, but echo the next poster's mention of it again. The .44's a better HD choice, but otherwise the good ol" .30-30's been doing all-around duty for over 100 years now. Don't listen to those with magnumitis or "it's worthless at 600 yards" types who pooh pooh the round. It's very effective for what it's intended range/duty is. It all depends on your quarry and conditions... like everything else, a shooter's "got to know its limitations." If you're expecting a .45-70 on moose in heavy timber or .270 far away, it ain't those. But it's a darn good all 'rounder under 200, which is where most folks' shooting is (or for many should be!) done anyway. Love my old Winchesters chambered in this great round, but the Marlin 336 is a good one also. In fact, hard to beat.
