Ok, so 2d6 Alchemist's Fire +2d6+12 sneak attack damage deals (3.5*4+12)*1.5 fire damage to a white dragon, which is an average of 39 damage. Assuming you get to throw 3 of them, that's 117 points of damage.
So after the initial assault, the dragon is kinda on fire. Each flask deals 2d6 fire damage on the subsequent round (6d6 total) unless the dragon takes a full round action to extinguish it. Assuming he does not in favor of an attack, he will take 3.5*6*1.5 = 31.5 points of damage, on average.
If he uses a breath weapon, the Swordsage will likely succeed on the reflex save. If he attacks, he will likely only get one attack off due to the distance between the two combatants. His melee attack bonus is enough to hit, though the Smoking weapon does offer concealment. The Child of Shadow stance, if used after the last flask is thrown, provides better concealment.
At this point, I'm kinda out of ideas. I guess engaging in melee or continuing to throw flasks of improved alchemist's fire at it is what comes next?
You could just hide again in you smoking weapon concealment? If you are optimised for it you can bring your hide check to more than +44, then you just have to have 10ft or more between you and the dragon to get a better than average chance of hiding.
@Twilight Wyrm, you calculate damage wrong, the average of 1d6 is 3.5 not 3, and 1d10 is 5.5 not 5. This makes your damage significantly higher.
If you think about it you actually are, if only slightly. Any steps require incremental raising, running would require restrained flight akin to hovering. I say this, because a running person does not move forward in a static line, such as a person under the effect of the Overland Flight spell presumably would. You are moving up and down, in motions you would need considerable control over your flight to replicate in mid air. This coupled with the other problems you bring up would seem to indicate a very low rate of success for such a tactic.
This requires at least more than 6 int to realise, and a pretty good eyesight to see that your opponents are not running correctly across a bridge in a dimly lit cavern. Also it probably requires a spellcraft check to know that there are things you can pass through while others fly above them. I would say this is a fairly good chance of succeeding, which means it comes down to what "interacting with" means.
Assuming the dragon is not inexplicably positioning itself to all vitals are out of sneak attack range for someone on the ground. (Since combat has not yet started, and as far as the dragon knows, no one is there (Given the meager 12 Cha available to Mature Adult White Dragons, their spells will be of limited use in this regard)
The dragon has a permanent
Spider Climb on all icy surfaces, assuming this white dragon is the standard kind who makes it lair in glaciers or icefilled caverns. It could simply be sleeping on the ceiling (and why wouldn't it).
Apart from that, I like your build and feel it would add as much as a tier 3 should to an adventuring party.
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I personally prefer the Warblade as a much better Tier 3 character. True his sneaky skills aren't that awesome, and need levels in factotum to really kick off. On the other hand: His offense is better than the swordsage, higher BAB, more focus on strength, and a bunch of maneuvers more suited for direct damage makes the warblade come out ahead. His defense is also better than the swordsage. He has the same substitute Concentration for Saves, but also has Iron Heart Surge, which is just crazy good defense.
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Since the supercharge, the TWF, and the Debuffing Swordsage has already been covered, I think I'll make one that focuses around Concentration checks and the nice Diamond Mind maneuvers. Coming shortly.