I'm right handed but have always shot long guns left handed due to my left eye being dominate and my right eye being slightly "lazy". This never allowed me to focus on a sight and target for any extended amount of time when shooting right handed. Needless to say, over the years I became very proficient with the right handed bolts often being able to take two deer running at the same time. It wasn't until I bought a .300 Win Mag Browning Stainless Stalker in a left hand version that I actually shot a dedicated left hand rifle and once I broke myself of the habit of bring my right hand back to work the bolt, I couldn't believe how much quicker it was for follow up shots. A left handed rifle makes all the difference for left handed shooters when shots are made off-hand or in a non-controlled situation unlike what you find when shooting off bags or using a bipod. Since you already have some left handed rifles, you're aware of this fact...that was more of a tidbit for everyone else.
Initially, I thought the WSM was nothing more than a fad and never had any desire of getting one...until Winchester introduced their Featherweights in a left hand version which were only chambered in the WSM calibers. Always having a love affair with the styling of the Featherweight stock as well as the Model 70 action gave me no choice but to finally get a WSM, like it or not. After doing much research, I decided on a .270 WSM as it offered the greatest performance difference from it's longer caliber's brother. I don't reload so I don't know what reloading manual the other member is referencing but ballistic charts on the Federal and Winchester show the .270 WSM thumping the .270 across the board and actually being more in line with the 7mm Remington Mag giving it a toe-to-toe fight and often slightly edging it out in the same weight bullets. For me, I'd pick the .270 WSM over the .300 WSM as the latter really brings nothing more to the table other than a shorter action and since I have a .300 Win Mag, there's really no reason for me.
I have roughly 200 rounds through my Featherweight and it adores the 140 gr. Nosler Accubond offering from Federal which I'm able to get MOA and more often than not, Sub-MOA groups with. I've probably taken close to a dozen deer with this caliber and every deer I've shot hasn't gone more than 50 yards with most dropping right in their tracks. My longest shot has been just slightly under 400 yards on a doe and she only went 40 yards before the double lung shot took it's toll. I can't say enough good things about the .270 WSM and couldn't be happier that Winchester shoved this offering down my throat with the Featherweight line. Had they offered standard calibers along with WSM's, I would have picked up one of those due to my initial thoughts on the WSM offerings and I would have never known what I was missing. I believe the .270 & .300 WSM are here to stay (unlike the other initial ones and the WSSM offerings) plus I think the .325 WSM will make it as well but time will tell on that one.
The main drawback is availability of ammo along with the cost of it. You're easily pushing $2.50 - $3.00 a round shooting the premium stuff and unless you buy online, you're very limited on the selection off the shelf. This is no big deal if you plan ahead but should something happen during mid-season or right before opening day, you're going to be screwed if you shoot something like the Federal 140 gr. Accubond load I do. Then there is the possibility of that load being discontinued (which Federal did this year) forcing you to find an alternate source or complete bullet change all together. Luckily for me, Winchester still offers it along with Nosler in their Trophy Grade ammo line but that one costs more and neither one is readily available at stores around here. My Featherweight doesn't group as well with the 130 gr. offerings but you may not have that issue with whatever rifle you choose. The other factor is the WSM calibers don't feed as smoothly as the standard offerings. It's not that they're a bear to get chambered but there is a noticeable difference at least that's been my experience.
For rifle choices, either one will serve you well but I'm going to give the nod to the Browning X-Bolt. As a matter of fact, I just came from picking up a LH X-Bolt Medallion in a .243 a few minutes ago for my daughter who like her old man, is right handed but left eye dominant. Of course I haven't shot it yet but it has some extremely cool features but like the Tikka, both are extremely compatible in features so it really is 6 of one, a half dozen of the other. I personally like the top tang safety and shorter bolt throw on the Browning as well as the oversized trigger guard. The bolt release button is a nice feature allowing you to un-chamber a round with the gun on safety. I was a little concerned with this but after messing around with it, it's a designed so you have to deliberately manipulate it so there's no chance of it accidently happening in the field due to it getting snagged or bumped. The big plus to the X-Bolt is the magazine and the way it feeds. The way the magazine is designed with the rotary type feeding, it lines the cartridge up straight with the chamber as opposed to being slightly offset like most other guns. This in itself would make a big difference in smoothing up the feeding of the WSM cartridges. As far as the magazine being polymer, that doesn't bother me as reliability and durability of PMags have shown polymer to hold up just fine. Of course, I'm a big Browning fan and the fit and finish on this X-Bolt is like all my other Browning's...SUPERB!
To make a long response a little longer, my pick would be the X-Bolt in the .270 WSM caliber as I'm sure it will bring you many, many years of enjoyment and you'll love the performance of this specific round.