Your truck can handle the towing if it is in good mechanical shape.
I have a 2007 Ford F350 4x4, single rear axle, 6.0 turbodiesel with 225,000 miles on it. It would be the same as the F250, but has a 1 ton rear suspension vs the 3/4th ton in the 250. I pull a 35' 8,000 pound travel trailer. By the time I have it loaded, I am pulling about 11,000 pounds. The truck can easily pull the load and hills don't have any problems. I have a 1200 pound weight transfer bar system that was set up by the dealer. My truck has had a lot of maintenance done to it to keep it up. Diesels take more maintenance. An issue with diesels is that a lot of owners don't keep up on maintenance which kills performance.
Your truck has the capacity pull your loaded trailer. My recommendation is to have a good diesel mechanic go over it to see what repairs it needs.
I bought my F350 used for 11,000 and it was in bad shape. In the last three years I spent around $15,000 on it of which $10,000 was mechanical. It is not new, but can do what a new one can and I have not spent $50,000 to $75,000 on a new one.
