Hi Milk, welcome to the club.
Definitely, changing the rear axle will improve the towing ability. What does it cost and is it worth it, I can't answer that.
But, a shorter ratio is not the complete solution, is your truck equipped with the factory towing package (and, I'm not talking about the steel receiver on the rear of the truck)? Read this thread, and maybe search for others:
http://www.gmtruckclub.com/forum/showthread.php/127694-2013-Silverado-1500-factory-hitch
Your truck should also have the 6 speed auto transmission. When I tow a heavy trailer, I use the M selection (instead of D), I do not let the tranny shift into 6th gear, I use 5th on level, or on down hills and, I manually shift (using the +/- switch) into 4th on hills. If the hill is real steep, I'll downshift into 3rd.
You also have the TOW mode button on the end of the shifter arm. Using the TOW mode, the transmission upshifts are delayed.