Found a nice write-up specifically for killing the dragons as Axton:

http://forums.gearboxsoftware.com/showthread.php?t=315129

I don't recomend you try Shamfleet (since that forces the use of elemental damage). Otherwise it looks good.

I already had all of the rifles it mentions farmed so I made copies for you guys.
We can still use the non-elemental infinities with the Sherif's badge and Fire Bee but these will make pulling the right dragon at the right time much easier.

Here is the Axton build I came up with that avoids all kill skills and Turret abilities while also giving you a ton of survivability and includes every other damage boosting skill you can get:

http://bl2skills.com/commando.html#05050051005500500515055055000050

I'm actually a bit jealous because the maya skill tree that avoids pgaselock abilities, kill skills, and elemental abilities is kind of terrible:

http://bl2skills.com/siren.html#550051050051531155000550500500

I also made new copies of the non-elemental Infinity, Sandhawk and Explosive Kiss of Death in case anyone sold theirs. We should be able to rock this fight!!!