Spec Specific Azerite Traits
Test of Might
Test of Might is the best stackable trait for single target and 2 target cleave. It is also good for sustained AoE. It does not work well in most burst AoE situations however, since you need to spend rage to get the benefit and not be bladestorming.
Crushing Assault is a very powerful to have 1 of this trait, and it’s OK to have more than 1.
Gathering Storm is very good on burst AoE and mediocre for single target.
Lord of War
Lord of War is very good on burst AoE and mediocre single target.
Seismic Wave is bad on single target and very strong on sustained AoE.
Striking the Anvil
Azerite traits rotation changes
Test of Might would have you Bladestorm during the Test of Might window instead of during Colossus Smash, since you need to spend rage during Colossus Smash with this trait. The Test of Might buff might also be stronger than the 30% damage from Colossus Smash if you have 3x Test of Might.
The other traits don’t really change the rotation.
Azerite Trait Recommendations
What traits you should go for is pretty obvious for Arms, and it depends very much on the situation. For raiding you will go single target most of the time and so 3x Test of Might is very strong. For m+ you will often go for burst AoE, and in that case you would try to get as many Lord of War and Gathering Storms as you can. You could also have a sustained AoE setup for m+ which would not hinder your single target DPS much compared to burst AoE traits, but it would require mobs to live for 20+ seconds at least so that you can benefit from Test of Might.
Overwhelming Power is an incredibly strong inner trait for Arms.
I do not recommend going for a BiS setup from m+ gear and instead save your residuum for the upcoming 8.2 patch.
Sim your character to see what is the best setup from what you currently have and potential upgrades from the raid.