Can f-150 tow Winnebago One dry wt. 6580?

First thing you need to do is to look at the yellow stick on the driver's door and see what weights are listed. Once you have those numbers, get the numbers on the trailer - specifically dry weight and carrying capacity along with the tongue weight.

Those numbers have much more to do with whether you can safely tow the trailer than the overall ability to "pull" it.

I can probably PULL my fifth wheel with an old Chev S-10 pickup, but I sure can't safely tow it.

You will get a lot of "OPINIONS" on here, but just make sure your numbers are within what the manufacturer of your vehicle says are safe limits. There will also be a lot of talk about being able to stop the trailer, but the trailer does have it's own brakes. That being said, your vehicle should be stout enough to stop it on it's own in an emergency situation.

After you get those numbers, you can feel confident about your ability to safely tow or not - no matter what people on a forum tell you.

I did not know you could get a V6 with a heavy duty tow package. I had a similar truck with a manual transmission and I would not have been able to tow that much weight.
Is your truck an automatic?
Are sure about the heavy duty tow package?

You can pull it, but truck ain't gonna like it! Your truck is rated to tow 5,400-5,800, depending if 2WD or 4WD. Even in good shape, your truck is closing in on 20yrs old.

Good luck,
Jerry