Google "trailer life tow guides" and look up the PDF for the year of your truck. You have the 3.92 gears which is a plus, but there are other considerations to help determine tow capacity. There's also how you drive and where you drive. What weight distribution kit you have.
I have a '13 quad cab, with 5.7 & 3.92 axle. My tow capacity is 10,050. Trailer is a '18 Keystone Springdale at 7,000 lbs. It's 33ft long. I have a Reese Dual Cam hitch. Once I got it dialed in, it tows excellent. Passing 18 wheelers don't really bother the rig. I don't have sway. I tow all over NY mostly sticking to 63-65 mph. Never above 70. I keep the tow haul mode on and watch my temperatures. Sometimes we go camping and the wife tows the boat with her minivan. I'll transfer weight there. Last season I put some Goodyear Endurance tires on and had them balanced. It makes the ride a smoother and betters handling. The 1500 does a great job and I'm confident enough in it to let my wife drive it.
I'd upgrade to a 2500 if I went out west in the mountains or take it all the way to Florida. I do neither. I personally feel you'll do fine speaking from experience in my truck. The weight distribution kit is key. Don't go cheap. If the trailer place installs it and it doesn't tow right, take it back for an adjustment. I had to.