Hi Freeheel, welcome to the forum!
You have a beautiful trailer there. Your truck should be fine to tow it, as long as you have the right equipment.
The Reese hitch is a good start. As Inland Andy pointed out you need to set it up correctly and have the right accessories. Namely the right weight of spring bars and the best anti sway setup.
Here is a link showing the various models of hitch.
Weight Distribution & Sway on Sale!
You will need the 600lb spring bars and Dual Cam Sway Control if you do not already have them.
You do not need new springs or shocks. Maybe shocks if the old ones are worn out.
For towing, you might consider adding a transmission cooler and changing to synthetic oil in the engine, trans and differential. Just a suggestion in the interest of reliability and long life.
The weight distributing hitch balances the load on all 4 wheels of the truck plus the trailer axle. The key point is getting the hitch set up properly. Make sure the ball height is correct. Then lift up on the spring bars until the whole rig is level. For fine tuning, go to a weigh scale and weigh the front end, rear end, and trailer separately and fine tune.
There have been other threads on setting up your rig. Once you have it figured out you just hitch up the same way every time and your rig will tow like a dream.
One other tip. If you find you need more power in the mountains, you might consider low profile tires. This has the same effect as lower gears in the rear axle but is easier and cheaper. Also makes the truck more stable in side sway situations.