Well, 1d6+3 = 6.5 average, while 1d12+10 = 16.5. So the Greataxe completely blows away your total combined damage even with Improved Two Weapon Fighting, let alone when you're making a single attack. Whether a shield bonus to AC is worth it is a different question, but that's how things lie with damage. I would guess that in most situations a +2AC isn't going to help nearly as much as taking down the opposition much faster. Since you've already put so many feats into it, I would start looking for anything you can find to make your two-weapon fighting work better for you. Though for an already-built Ranger in a low-magic game I'm not sure how much there is to do about it. Maybe if you were allowed to retrain a feat...
Oh, and of course Favored Enemy helps, but that's very situational until you can cast level 3 spells.
It seems like there's something wrong with your math since the shortsword and the greataxe both have the same chance to hit. Let's analyze the situation where the enemy AC is 19. As you said, we won't get into crit threats.
Mw. Greataxe: Average damage per hit is 16.5. The +8 attack is 50% likely to hit (11 or up) and +3 attack is 25% likely to hit (16 or up). That's 75% altogether, so that's 12.375 average damage.
Mw. shortsword and mw. heavy shield: Average damage for both weapons is 6.5 since you've got Double Slice and presumably a spiked shield. The chance to hit is the same, but there are twice as many attacks, so multiply by 1.5 to get 9.75.
+2 battleaxe (maybe): If this is really a +2 axe then at +9 to hit with Power Attack it would do 1d8+9 damage for an average of 13.5 per hit. Your percentage would go up to 85% for 11.475 damage.
If that's really a +2 axe then using it with a shield for defense it might be your best bet for the moment. I'd probably go with the TWF since that seems to be what your PC is built for though. Incidentally, I think your average DPR when using Power Attack with TWF would end up being around 10.45. You'd also have the chance of adding a lot of extra damage against favored enemies and set yourself up to take the Shield Slam feat soon. I highly advise Shield Slam.
Remember, you can always attack with just one weapon when you need greater accuracy, and you could even drop your shortsword and use the shield two-handed if you needed a little extra damage and accuracy. Don't give up that AC bonus and the chance to push foes around.
I think he was counting using Power Attack with the greataxe but not with the TWF, since hitting was problematic.
Power Attack is an even bigger deal when playing a low-magic game, because you don't have extra damage from enhanced strength or weapons; unfortunately Power Attack isn't at it's best with two weapon combat, especially if you're struggling to hit. Using it with your shortsword and shield would give you around 60%/30% more damage, but if you can't land hits...
The feat-swap I was thinking of would be to drop Improved Shield Bash and pick up Improved Unarmed Strike, using axe and unarmed strike and just carrying the shield; but I was going to suggest it partly because of Magic Fang, which apparently is not on the table. Instead, you could also just pick up some masterwork armor spikes to keep TWF to a -2/-2 while carrying your shield, which would also render shield bash unnecessary.
EDIT: Shield Slam is indeed nice, but note that it calculates your Bull Rush based on your attack roll. If you're having issues with a low attack roll, shield slam is going to require a good deal of luck to do much at all.
Are your fights pretty varied? Sometimes 1 BBEG, sometimes a few mid-level challenges, sometimes a whole bunch of weak mooks?
If so, use the greataxe against the BBEG and the 2-weapon fighting against all the weak mooks.
Yeah, I actually figured out the -2 for TWF vs Power Attack but then forgot to delete the first sentence - sorry about that. Anyhow, at least by my calculations the TWF damage should go up slightly with Power Attack vs AC 19.
In theory you can make TWF attacks with either weapon first, so you could stab a couple of times with your shortsword before using Shield Slam to push them away. It is also great fun for pushing enemies off cliffs or bouncing them into walls, trees, etc so they fall prone. Eventually you can also pick up Shield Master, which I guess might make a big difference in a lower powered game.
One thing I'm wondering..have you thought of going with the axe and a light shield? You can still bash with it, although indeed you are taking a -1 to your armour class.