Most of us have been there
Free range pigs sounds wonderful until you realise that outdoor pigs on wet cold mud up to their bellies are not happy pigs. And indeed, the welfare codes disallow it; there is provision in all the free range regs, including organic standards, for pigs to be housed indoors on straw when such conditions prevail.
But as I said, most of us have to learn this the way you are! So after the winter, think about whether you will have pigs over winter again, and if so, make other arrangements.
For this year, the first thing is to make sure their beds are dry. Don't assume, get your hands onto the straw and feel it. Straw bedding gets soaked, and stays wet and cold, very quickly in these conditions. With just two adult pigs in similar conditions, I was putting three or four slices of straw (from a small bale) into the ark every day. My routine was to, each day, fork out the wet, cold stuff from the entrance and use it to build up a mat outside the entrance, which gradually became sufficient to help dry off their legs and bellies as they went back to the ark. Then throw in the fresh slices for them to use as they wished. They'd build a windbreak wall inside the entrance and make their bed behind that.
If I had a reshy bale of hay or any other similar waste, I'd add that to the apron too.
My two quickly realised what I was doing, and its benefits, and took over putting out the wet straw onto the apron
When they are wet and cold they won't spend much time rootling as long as they have plenty to eat. So feed them well, add a little warm water to their drinking water if it's frosty, and as long as their beds are warm and dry they will probably spend most of their time in there, apart from when they are eating.
If the mud is higher than their knees, and definitely if it's up to their bellies, you would really prefer that they don't have to spend a lot of time in it. So feed them as near to the ark as is practical, and maybe make another straw-y apron where you feed them and put a thick, heavy rubber stable mat on top of that, under their troughs, so they can stand on dry rubber to eat. Yes they would rootle all that away in no time if they felt so inclined, but they won't want to be out in the wet cold mud up to their bellies longer than they have to, so if they are given plenty to eat (and the food doesn't get into the mud or straw), they will probably not rootle at all.
You may find they start to poo and pee nearer to the ark, or if the ark is very large, even in the ark. If they mess in the ark, that's telling you they are very, very uncomfortable outside. Help them stay clean by making another straw-y path to their toilet area.
I expect you could use bark chips or anything of that type along with straw and waste hay etc to make a few drier areas. Personally I would not do more than the ark entrance, and if necessary, routes to and from toilet, feed area and drinking area, and under a mat in the feeding area.