I don't see how $8.00 is a low price for a 5/8 nut. You can buy 5/8 nuts at Home Depot for 6o cents.
For common uses, a nut is a nut. What you're most likely to get at Home Depot is a common nut for a common use. That means that what you put it on isn't going to wiggle, flex, rattle, or make a lot of noise.
For repairs on something like a mower, where there's likely to be a fair amount of vibration and variable loading, you will generally be much better off going to an auto parts store for your nuts and bolts. Hardly anything "automotive" will use fasteners lower than SAE Grade 3, and fasteners appropriate for use directly on engine components should usually be Grade 5. Some chassis and suspension fasteners are commonly Grade 7. (There are similar ratings for metric fasteners, and for other than SAE standard fasteners of kinds more common in the UK and elsewhere.)
While there are very significant differences in the "strength" of the various classes, the more important difference in many cases is that the "graded" fasteners generally will use UNC/UNF threads, rather than NC/NF ones. The "U" standard requires radiused thread roots that are much more resistant to fatigue failure, and they must be inspected and certifiable as to material quality and treatment.
I've had "common" grade nuts split just from being screwed onto the bolt/shaft at torque-wrench measured torque below their "rated strengths," simply because there is no "quality control" that can be relied on.
That said, a 5/8-UNC SAE Grade 3 nut, bought off the shelf, would likely be closer to $2 than to $8. The Sears price is no bargain, unless it's in some way rather "special" (left-hand thread, perhaps?) but is perhaps "realistic" for a specific part that has to be stocked and handled at low turnover rates.
The most common cause for "having a loose screw" is failure to install it to proper torque. If the machine shakes, rattles, or rolls, using a properly calibrated torque wrench and getting it to the correct tightness when it's first installed is critical. Too loose, and the threads will be unloaded, and the fastener will back off. Too tight, and the fastener will be stretched (or cracked/broken), which makes it too loose, and the fastener will back off.
The next most common error is putting dirty parts together (partly because any significant dirt makes it impossible to get an accurate torque). It's not (usually) too essential that you sterilize your nuts when working on a lawnmower, but you should at least shake the gravel off before starting to screw.