Here's an example where I had removed the solid 6x8 wood posts and replaced them with steel adjustable columns. I had fabricated them myself and secured them to the footing piers with concrete anchor bolts ... with the blessing of the building inspector.
I did this job ten years ago as part of a partial cellar excavation, on a house such as yours, of similar size.
The town made me hire a civil engineer (Yellow Pages, "Engineers, Civil) to draw up plans of how to go about it before they would give me a permit.
Thankfully, they allowed me and my crew to do the work; the engineer cost $500, but he was good & visited the site several times & offered great advice.
The first step is to get the floors & beams level & plumb; as you noted, take a nylon string & tightly stretch it the length of each floor & along the lower beams, & use a little line level to get the string plumb.
You should immediately be able to tell how many inches out of line not only the basement beam is, but also the floors & attic above.
The beams of a house are like the old bones of an aged elephant; they often creak & moan when jacked back into place; don't expect perfect symmetry.
It's ok if the plumb is a little off after a month of jacking, as long as it's not too noticeable on the floors.
Also, go out in the street or your yard & visually check the horizontal line along the roof peak; there should be very little or no apparent bend in it.
The adjustable posts should be placed on firm footings or heavy rocks every 4' to 6' and cranked GRADUALLY a few turns a day to get the beam & floors level.
This will take a month or so if done properly, so the ceiling & wall plaster doesn't crack.
Remember the building's been slowly sagging for years & has to be brought back to plumb gradually.
The adjustable posts are only $10-$15 each & come in different lengths; they can be cut with the same circular saw blade you're using on the cast iron sewer pipes if you need a short one to use in the sub-basement.
Individual concrete pads are rejected in favor of a single steel-reinforced strip concrete 2' wide X 1' deep footing running the entire length of the house along the entire length of the sag.
Steel 3/4" rebar (at least 4 continuous pieces) is used in the new footing as the reinforcement.
Since the adjustable posts are in the way of pouring the footing, one or two posts are temporarily removed & the footing is poured in 6' increments with the rebar sticking 2 feet out the end of the pour with a keyway so the subsequent footing pour CAN BE TIED TO THE PREVIOUS POUR, & interlocks with the keyway and the steel rebar.
This interlocking tying in of each steel-reinforced subsequent footing pour is the key to a firm footing that won't crack or sag.
If there is a cellar window near the street this procedure is much easier if a concrete ready mix truck can simply extend his chute into the cellar & pour directly into the footing forms.
The removed adjustable posts are put back in position after the concrete cures (2 days), & the next footing pour using the same subsequent method is executed.
The new footing is then built up to the existing beam using solid concrete block masonry.
A sill of pressure treated 2 X 6 or 2 X 8 is used between the new concrete block & the wood beam.
Hi Jack: Is that keyway put in horizontally or vertically when when you pour the footing? Thanks.