Get a used axle assembly from a junked H3....they're easy to find and cost around $500 to your door. I've bought several this way, and its far faster and cheaper than trying to repair the one thats currently in your truck. Obviously, you need to make sure you get the correct gearing (4.10 or 4.56). It would be nice to find an 09-10 axle, to get the cast-iron version. Additionally...it would be nice to find an ADV PAK version too, so you could get the locker upgrade. However, since you're probably grinding teeth inside it...you're in a bit of a time crunch (no pun).
Try one of the nation-wide salvage websites like Schram. I've bought two front axles from them...always got fast delivery and good prices. Also try Wayne Auto Parts in Ohio...they've got 23 Hummer's they're currently parting out. Realize ahead of time that these guys rarely know what an ADV PAK is...or what year its from. Their computers simply say
"H3 Front Axle 4.56 gear 2006-2010". Ask for a picture ahead of time, if you have any doubts. They will usually email you a shot of the exact part so you can see what you're getting. The CI version will be painted black. The alum version will be raw silver finish. If you can't find one let me know...I know some top men.
Swapping a front axle is cake. Just pull the brakes, hubs, half-shafts, and front driveshaft. Then drop the lower crossmember. Three bolts hold the whole thing to the truck. While you got it out...put some REAL bushings in the outboard mounts. These will help keep it from flopping around under there. Nice product here, and sorely needed on any H3.....
REAL BUSHINGS