Well I just killed my first dragon outside of White run. The Jarl rewards me with an "Axe of Whiterun". However, it never shows
up in my inventory. I never receive it. Nor does it flash across the top of the screen saying, "added to your inventory" like other items do.
why dont I get it?

That sounds like a bug.

The good news is that it's not useful anyway, so youre not missing anything. Mine is collecting dust in a chest, and despite what the Jarl says, nobody treats you differently when you wield it. They just think its an axe, and you'll probably find or craft a better one soon anyway.

Thats why you should save before stuff like this.

Its not useless, it gets an enchantment dependent on your level, the axe type is also dependent on your level. When I got mine it was worth 5000+ gold.

I have a save before killing the dragon and also entering the Jarl's Hall afterwards. Both saves, I'm near carrrying
capacity. What should I do?

On that latter save, just exit the hall, wait until 8 or 10 AM, and sell some stuff at the shops. Surely, some of that stuff you're carrying is just vendorable loot.

You got an axe? o.0

He gave me a Sword of Whiterun. I wonder what effects what he gives you...

Well apparently I'm the only one who received nothing. I tried to pick pocket him to see what he was carrying and he didn't have it on him.
What should I try?

Interesting mine was an axe too and not a sword. I guess the game randomly generates the weapon type. To honest, it's not that great. It's a d12 weapon that does some fire damage. You can pick up better loot!

I have both an 2-hand Axe of Whiterun, and a 1h sword of Whiterun.

I got the axe by helping the jarl as my first city to get a house, that got me the axe.

Then, I completed the Stormcloak quests, eventually overthrowing the Jarl and putting one of the Grey-manes in charge, I spoke to him after the coup d'etat and he said I was such a great guy for Whiterun and gave me the sword.

UPDATE; item finally received.

Okay so after restarting just prior to killing the dragon outside Whiterun, I was able to get the Axe-of-Whiterun from the Jarl. Interestingly,
when I approached him THIS TIME I could see that he was holding the axe at his side. For some reason he didn't have the axe on his possession
the first time(which could explain why I wasn't receiving it).