Since you already shoot left-handed, I'd recommend switching to LH rifles. I'm unfortunate enough to be left-handed, but fortunate enough to also be left eye dominant. All of my bolt action rifles and my most used shotguns are left-handed now. It takes a little getting used to after using right handed guns, but I found it to be worth it. That said, selection is rather poor for lefty rifles.
After making do with what I could find for quite a few years, I finally just bit the bullet and started doing custom builds. You didn't mention a budget, but if you're willing to forego controlled round feed and walnut, a nice lefty Remclone can be built for under $3k. I'm about $2.5k deep into my latest build with a Pure Precision Summit action, B&C stock, McGowan Barrel, and Triggertech Primary. I'll probably get to about $3k once I add NECG express sights. I shopped sales and have about $1950 into components. There were a few gunsmithing challenges and some modifications required that cost me the additional $550.
As far as currently available factory options, the Ruger African or Guide Gun are nice rifles, available left-handed, and quite affordable. They're chambered in .375 Ruger, of course, which does limit ammo availability as compared to the H&H. Hornady and Buffalo Bore offer .375 Ruger ammo.
The Winchester Safari Express and CZ 550 were also available for a short time in left-handed .375 H&H versions, but both are now discontinued and when you find one you'll pay rather dearly for it. At that point, the more basic Blaser models start to make a lot of sense. I have no personal experience with them, but many here recommend them highly.
I stumbled across a great deal on the African and Alaskan rifles many years ago, so I bought one of each, both in .375 Ruger, with the intention of rebarreling one. For various reasons, that never happened, but both rifles have proven to be accurate and totally reliable. I do recommend that they be glass bedded and have a better recoil pad installed. They're a bit rough out of the box, but slick up with some use.