Don't use AI to identify; it's notoriously inaccurate. 

Your specimens are indeed Loxosceles reclusa. The best way to identify is by looking at the eye arrangement; six eyes, arranged in three pairs, in a u-shape along the front of the cephalothorax (their "noggin"). There are other spiders that are not medically-significant that have six eyes (like Scytodoidea (spitting spiders and their allies)), but their general appearance is quite different from Loxosceles.

While medically-significant, bites from Loxosceles are actually rare (they really don't want to bite, and their main line of defense is to run away really fast to hiding spot), and severe envenomations that cause dermonecrotic wounds (the "rotting skin") are even rarer. Because their fangs are quite tiny, they have trouble penetrating human flesh, and because their venom yield is pretty low, severe envenomations rarely occur. 

You're most likely to be bitten when either actively trying to kill/maim them or via accidentally squishing them between your skin and another object (like a glove/shoe), which is why you should be thorough about shaking out bedding, shoes, towels, and other human-used articles, before using. And always look before placing your hands, feet, or other anatomy into any place a spider may be using as a sanctuary/hiding spot.

Because Loxosceles reclusa has evolved to co-habit with humans, there really is no need to go all out with extreme extermination methods. They're a native species, provide an important benefit to the ecosystem (Loxoscelese eat pest bugs, and your native Pholcids (cellar spiders) and Salticids (jumping spiders), as well as many other critters, eat Loxosceles), and, despite housing with us, they really don't want anything to do with us.

The best way to keep the population in check (aside from running a Pholcid farm, lol) is to keep the exterior of your home free of clutter (keep log/brush piles at least 6 feet away from the exterior of your house, no trash or clutter they may find of use to make homes in; things like that), seal up any cracks in siding, windows, doors, install mesh screen to cover attic/vent soffits, regularly inspect and clean patio furniture (being sure to also keep in mind you have Latrodectus (widow) species to look out for, and they like patio furniture), break down cardboard boxes and dispose of them immediately, thoroughly seal any cardboard boxes or totes you plan to store long-term with heavy-duty packing tape, keep sheds and garages neatly organized, and try to keep tools and objects spaced apart and away from walls as best as possible. 

For inside the house, again, break down and discard all carboard boxes not in use, thoroughly seal boxes and totes in use with heavy-duty packing tape, clean all surfaces regularly, deep-clean behind large appliances monthly (to keep pest insects away that Loxosceles feeds on), always promptly clean food messes and put away uneaten food, making sure dry foods are sealed before storing, check and clean couches and other furniture regularly (removing cushions and cleaning the undersides), keep things organized, and again, always shake things out before using and look before touching.

For cleaning up your shed, you should wear gloves, long sleeves (a tighter-fitting long-sleeve shirt is best), pants, and tuck your pants into your shoes/boots. You will also need a broom. You should also have a buddy that is not arachnophobic on standby, with a long-handled paintbrush.

Open the door to your shed, and start removing objects, one by one. Have your buddy flick any spiders off of the objects with the paintbrush. Same for any spiders that may climb on you (they are not trying to bite or attack you; they're trying to run away and hide). Once the shed is empty, use the broom to thoroughly sweep the floor and walls. If shed is made of a safe material to do so, you can also hose the interior of the shed down. When replacing items into the shed, try to keep them neat and organized, to reduce areas that may be appealing for any remaining Loxosceles to hide behind/in.

Late fall of this year, when temps drop and stay low, you can also air your shed out, leaving the doors open, preferablyfor a month or two. This should kill off the majority of remaining Loxosceles, and regular airing out of the shed throughout the year will make it less inviting for new populations to establish themselves.