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The 5.0L engine will have the power to pull the weight but can the truck carry the weight you'll be asking it to carry?

I'm with ctbruce. The first spec you'll exceed will always be the payload number, then the rear axle weight. That was certainly my experience.

So my advice is to forget the "towing" capacity and concentrate on the "payload" capacity of your pickup.

If you end up towing 7000# actual weight with an F-150, make sure you upgrade your tires to LT tires. You might need to do air bags, too, but just remember that they are bandaids solutions and don't actually change the payload capacity of the truck. The axles, wheels, bearings and frame have not changed.

Good luck.