A few thoughts:
Why don't you take the hiddys out and leave the cozys as a place to hide and then when they are in them you come up from behind and scoop them up? I would just switch them over to cozys full time. You don't need to have two things for two actions being for sleep and pick up. It would help them tame quicker if they knew their only place also moves because then nothing is in their control. Then when you get them tamed add new places of cover, and/or fun for them later on. You keep hitting reset when you have to pull the huts up, because starting out their safety place is taken up right away. Then they run and you can't grab them.
Also, something to keep in mind about pigs is that while they are prey animals, and we have emotions that make us feel that being delicate with them is the best way to handle them, it actually is beneficial to make them have to deal and live with you. A lot of times you'll read on forums about how, like a light switch people's pigs suddenly liked them after months of hand feeding. Eventually all pigs surrender and give up fighting their owners, as if the owner is named top pig. Your top pig> top pig > second pig is the hierarchy lol. So you have to treat the situation as normal and that they need to live in your world, and not catering to them. You wrote that your talking resulted in mad purring from them as if you were interfering, when they need to deal with the sounds to learn how to live with you. If their hiding spot is a cozy which also lifts up, and they have to hear and see you non stop, eventually they say ok, I surrender. I have to deal with this human and food comes from them. There's a difference between intentional stress harming them, and simple stress of them having to accept to live with humans. So I think your doing a good job and don't be afraid to be in their space.