That truck will pull your trailer. The 12,700 LBS is the GVWR (gross vehicle weight rating) which is the amount the truck can weigh, whether it is loaded with two pallets of lawn grass, or the hitch load. If you pull your truck onto a scale with your trailer hitched to it, and the trailer is off the scale, the weight of the truck should be no more than 12,700 LBS.
The next weight that you did not mention can be found somewhere on the truck or owners manual is the GCVWR (gross combined vehicle weight rating). That number is the max that your truck and trailer can weigh in the travel configuration. This is the weight of the truck and trailer on the scale at the same time.
You did not say if the truck was a DRW or a SRW. If it is a dually then you probably have the correct tires on it. But if it is a SRW, you need to look at the rear tires and see what you have. Used trucks have a bad habit of having the wrong tires on them. So when you get your truck and your trailer hooked up, go get it weighed, weigh the front axle, then the front and rear axle, then the rear axle by itself, then the truck and trailer, then the trailer only all while hitched. This will give you all the information you need to determine if your truck is within specs. But when you get that trucks rear axle weighed with the trailer hitched, compare that weight to the weight rating of your tires. Like I said If you have a dually you are probably in good shape, but if you have a SRW, you need to watch the tires. I have never heard of a SRW with a 12700 lb GVWR so I'm thinking you have a dually.
You should be looking for a number between 24000 and 26000 lbs GVWR for that truck in a SRW, in DRW the number is probably 30,000 lbs. This data should be on your door post. If not its in your owners manual. Options move these numbers up and down, the addition of crew cabs, 4X4, axle ratios and other options can change these numbers.
Hope this helps.
Okay I need some honest answers!! Lol--My truck is rated for 12,700 lbs. 2004 F350 diesel.. Im looking at the Big Country 3070re which st ates a 14,000 full loaded weight caoacity. Is this a concern for me? Ive hot $10,000 going into the motor to make it reliable for just thus reason. No real mods but just bulletproofing the motor. It looks as though most fufth wheels I see out there are bigger than what I am looking at.