Well, what I would do is somewhat different than others. But that's normal for me.
I would get a 2 barreled pump or semiauto 20 or 12 guage, one long and one short. Reasons for it are obvious and you stated them in your OP - upland rabbit and birds, close range deer with slugs.
I consider your .44 Mag carbine to be redundant when you are going to have a short-barreled shotgun, but that's just me. If most of your available deer hunting is in brush, then either will work great for that kind of hunting, but if you are starting from no hunting guns, getting both at the same time is kind of redundant. Just wanting one is sufficient reason to get one, of course, but you have to recognize that the .44 isn't giving you any real increase in hunting capability when you will already be setting up a short barrel on the shotgun for slugs.
So, start with the 2 barrel package shotgun from Remington or Mossberg.
If the biggest game you will hunt now is deer, then pick a rifle that will give you that and coyotes and other varmints with little fuss. Get a .243, .260, 6.8, .257 Bob, .25-06, 7-08, or .270. Or something like that. You said you are avoiding getting a .30 cal at present due to panic buying drying up ammo. I would agree with that, with the exception of the .30-06, since that seems to be on the shelf everywhere around here, but that's probably a regional thing. It's not hunting season anymore here, so few people are thinking "buy a stack of that .30-06 for in case there's none of it in October." But the -06 is bigger than you need for deer and coyotes. If you just want one rifle for everything in North America, of course, then it's your Huckleberry.
Later on, as your hunting needs and finances increase, pick out your big rifle for that Africa or Alaska trip, and pick up that .44 carbine you're wanting.
My advice is to run your priorities for hunting guns like this: Just a shotgun w/2 barrels and necessary chokes, one good rifle that is adequate for your local game, and a .22, and you're set until you are going somewhere that requires something different, or until you've saved up enough to get stuff that is more of a 'want' than necessary to accomplish your 'needs'.
And of course, there is the muzzleloader if you are interested in hunting during one of those seasons, but you didn't mention that.