To do caliper measuring of an uncleaned dug iron ball, I use a hammer to gently tap the rust-crust off of some small spots on opposite sides of the ball.
It's best to measure a ball's diameter in at least three directions, because that how you tell if the ball is out-of-round. But seeing photo #3, which shows one side of your dug ball is already pretty clean, you can measure it in two directions (let's say, north-to-south and west-to-east) without having to do any encrustation removal. For the third measurement, clean a small spot directly opposite from the center of the clean side.
If you don't have access to a large-enough caliper (it jaws must fit across the ball), you can use what is called a Diameter-Tape. But the tape MUST be positioned exactly on the ball's "equator" to get accurate measurement. See one of Diameter-Tape in a photo (and info) here:
http://www.pochefamily.org/books/SolidShotEssentialsMod.html
Precision weighing can also tell whether a ball is at least "possibly" a cannonball or not. For example, the Ordnance Manual specifies that a civil war 12-Pounder caliber Solid Shot cannonball should weigh 12.25 pounds (12 lb 4 oz). I should mention that it can also weigh an ounce or two less, due to casting-flaw bubbles inside the 19th-Century cast iron. But if a solid (not hollow) ball weighs right-at 12.0 pounds, it is a Sports Shot-Put ball (from the 12-pound Weight Class of that sport.) For fairness in Sports competition, the Sports Shot-Put balls are very carefully manufactured to always weigh exactly 12-pounds... because it is unfair if the other team's ball weighs 12.0 pounds and your team's ball even a little bit less than theirs.
So, the very first thing I'd do with your solid (not hollow) ball is weigh it on a Postal Shipping scale. If its weight is more than an ounce or two different from any Solid balls in the Ordnance Mnual's charts, it cannot be a cannonball.
For your sake, I hope it is a cannonball. Based on comparing its size with your hand in the photo, it MIGHT be a civil war 6-Pounder caliber Solid Shot cannonball, which was specified to be 3.58-inches in diamter and weigh 6.1 pounds (6 pounds 1.6 ounces).
Do you know if there was ever a battle involving the use of cannons anywhere near where you found that ball?