Well I like where you're going, but I believe the main issues with his kit are his E.
Now don't get me wrong I love his dash. It has gotten me out of tough scrapes and lets me do the famous Shurima Shuffle. But that is the main issue with balancing him. It's like Kalista. She has insane mobility so they had to weaken everything else about her to compensate.
I love doing the shuffle, but in a real fight it usually isn't the best option. If I dare dive into the enemy team, I just handed my enemy a high DPS carry on a silver platter. Even if I get a kill or two out of the shuffle, I would do more good by sitting behind and attacking with my soldiers.
The Shuffleis honestly just a flashy play and a noob trap to be honest. His ult should be used for self peel. Now for his E.
An emperor stands behind his soldiers and fights through them, so he shouldn't be able to dash with his E.
This encourages the playstyle of standing back and summoning an army to fight for you. With this change we can give Azir back most of his power now that he will want to set up shop and wreck from one place. You get to him you beat him.
Now to compensate Azir players. Emperors do not only have soldiers, they have bodyguards. I propose that E summons a different sort of sand soldier, a bodyguard. They will work on an ammo system like sand soldiers, but they have a longer cool down and only one can exist at a time.
They can last at least a minute, and Azir can't move them. They will dutifully follow their emperor.
If an enemy gets too close, they will charge them, exploding into sand on impact and knocking them back, giving Azir a speed boost depending on the rank of the skill. This gives him an escape but doesn't make him impossible to catch. It gives him a fighting chance.
Now we can let his soldiers do separate damage, giving you a reason to spam soldiers.
His play style will be to create. Zone of control, a camp if you will. He will command his sold from there with bodyguards to defend from ambush.