I remember seeing a chart showing weapons that could be used as other weapons of different sizes. E.G. a medium shortsword is equal to a small longsword. Both do 1d6 damage and have a critical rating of x2 on 19-20. This would allow a small character to use a medium shortsword as a small longsword. (I think this is not RAW)
This sounds like it might have been referencing the 3.0 rules, which did something kind of like that. As you are playing 3.5, best to ignore it. A mis-sized weapon remains that type of weapon, just wielded differently.
Anyway, on to the meat of the question: yes, if you have proficiency in it, the dwarven waraxe can be a one-handed weapon. If it is one size too large for you, it becomes a two-handed weapon for you, and you take a −2 penalty on attack rolls with it. So at best, assuming you are a dwarf, you can take a −2 attack penalty to go from a 1d10 (average 5½) damage weapon to a 2d8 (average 9) weapon. If you are not a dwarf, this also costs a feat, or the penalty balloons to an untenable −6.
Trading −2 attack for +3½ damage isn’t really a great trade, seeing as Power Attack would get you +4 for the same −2 penalty and can go beyond that. Realistically, you don’t want to take any attack penalties you can avoid—Power Attack loves Shock Trooper because it eliminates the attack penalty, and likewise, anyone wielding an oversized weapon wants a way to do so without penalty.
There is a terrible feat for that, Monkey Grip, which halves the penalty. A feat is a deeply precious thing, far too great a cost to only halve the penalty. Far superior to that are powerful build or the strongarm bracers from Magic Item Compendium. The best ways to get powerful build is the goliath race from Races of Stone, but it has LA +1 and isn’t really worth it (though the goliath barbarian substitution levels in Races of Stone can tilt the needle in their favor). Meanwhile the strongarm bracers cost just 6,000 gp. Since you can only have one race, and strongarm bracers explicitly do not stack with powerful build, the bracers are the way to go. Spending 6,000 gp on a +2 bonus to attack is clearly worth it.
But then there is no point in using a dwarven waraxe: with strongarm bracers, you can wield weapons as if you were a size category larger to begin with. That means you can just use a large two-handed weapon: a large, 3d6 (average 10½) damage greataxe represents a +5 damage bonus instead of +3½.
You could also use the strongarm bracers to wield a huge dwarven waraxe as if you were large, i.e. with a −2 penalty. That would be 3d8 (average 13½) damage, another 3½ greater than the large greataxe. Again, though, −2 attack for +3½ damage isn’t a great trade, and since you’re already using the strongarm bracers, easy approaches to dealing with that penalty aren’t available. You could go with the Wield Oversized Weapon feat, but again, a feat is a huge cost—and now this needs two of them (since you need to have Monkey Grip). And even if you did, again you would want to go with the 4d6 (average 14) damage huge greataxe over the huge dwarven waraxe. Two feats for +4 damage is pretty awful, though.
But we can actually go even harder with this. You might go with a gargantuan greataxe, for 6d6 (average 21) damage, since Wield Oversize Weapon makes it simultaneously count as a size category smaller and a one tick “lighter” in effort, i.e. you can treat it as a huge one-handed weapon (that you can wield as a large two-handed weapon at a −2 penalty). Two feats for −2 attack and a +10½ damage is starting to look kind of good, though you definitely can do better.
In short: it seems to me that the best approach here is to use a pair of strongarm bracers, and to use a large greataxe, rather than a large dwarven waraxe. That, or if you really felt like you could take the penalty and wanted to squeeze the most out of this, go with Monkey Grip, Wield Oversized Weapon, and strongarm bracers to wield a gargantuan greataxe at a −2 penalty.
You’d still do vastly more damage with a lance, though, and abusing charge damage multipliers.