First off -- good luck.

You want "power", but are complaining about the slowness and small lift
height of your jacks. Those two facts are related, of course -- if it
went faster, you'd have less power. Laws of physics and all... But
renting 20T bottle jacks should be cheap.

I would be real hesitant to use railroad ties for anything in this
project, other than laid flat on the ground for a temporary base.
Especially if they are old, used, or weathered. Remember, these things
are designed to lay flat to take a load, not raised on end. Loaded
end-wise, they could split, splinter, or collapse. Use steel screw-type
columns, or decent wood 6x6 posts. Or better yet, use a 3'x3' stack of
4x4 lumber (see below).

There are some other things being neglected. Supposing you do get the
whole barn up (30''!) in the air, supported on lally columns, or
whatever. Just the slightest wind will tip your whole barn over. You
need horizontal (diagonal) bracing, and a lot of it.

The way a house-mover does it is to build several stacks of 4x4 or 6x6
lumber -- in the corners, along the walls, in the middle, etc. Each
square stack has two pieces laid down about 3' apart. The next layer
has two pieces laid 3' apart across the first layer, and so on, in a
tall stack. Takes a lot of lumber to do, but will give decent
horizontal bracing. Especially if you nail some diagonal bracing to the
stack. As you raise, you put another layer on at each 4'', and every
few layers bind the whole thing together with another diagonal brace.

A house mover would probably use large steel beams on top of the
built-up columns, to support the floor. You could just use 6x6 beams,
or something, though.

Finally, for the walls, I would probably go with poured concrete walls
and footers. It seems by far the easiest, and probably the cheapest as
well. A concrete contractor might be able to tell if the remaining
stone footer is good enough, and just put up frames and pour on top of
that. It take little labor, and be done in a matter of days.