The suggestions so far have been good ones, so let me add one more to your choices. I used to straighten old and sagging barns, repair bulging stone walls under barn banks, move barns, etc. I have a little bit of experience in what you're pondering and the barns around here in Amish country are the same style. For that matter, when I looked at your first pic I thought it was my wife's old barn.
What we did is use bottle jacks setting on a pyramid of shorter timbers placed crosswise to each other, one pair running N-S, the one above that E-W, until the pyramid was close to the beam we wanted to support. The top course of blocking was solid and double. The bottle jack sat on that, then a reverse pyramid was made of progressively longer pieces of wood until we had support under about 2 feet of the beam. The bottle jacks were slowy raised, taking the strain a little at at time.
In the situation you have, it looks like someone was a little eager about cleaning the barn and removed the dirt from around the rocks that the posts rest on. They may or may not have bumped a post with a piece of machinery. The rocks are probably not going to move much, it will take a pretty serious piece of equipment to move those things. I would consider digging out around the rock, forming up an area of suitable size, and pouring it level with the top of the rock with 6 bag mix concrete.
Then, if you anticipate keeping critters in there and are worried about the bottom of the posts rotting or being damaged during cleaning, cut the post off about a foot above the rock. Get a 5 or 6 gallon bucket and fit it around the post. You could cut the bottom out of it if you want, and maybe set some re-bar into holes drilled in the rock or your new concrete but sticking up into where the bucket will be. Put the bucket down on the rock/concrete and pour it full of concrete, making sure to shake, tap, hit, tamp, etc. to get all the air bubbles out. When it all sets up, you can leave the plastic or remove it at your pleasure, but either way, your posts will be connected to the base and each post will have a nice protective concrete base about 18 inches thick under it.
As has been said, those rocks aren't likely to move. You really won't have much of a frost problem unless you get water in the soil on the barn floor or have a spring in there or something. It'll freeze, but not as deeply as a bare field, plus any animals -- especially manure and bedding, will tend to keep it relatively warm.
Get to know your Amish neighbors if you have any and let them know what you want to do. They will appreciate the chance to help you out so you can return the favor if they need a phone, a ride into town, or whatever.