Step 1-you will need a design for the new foundation--It sounds like you plan on just putting some shims or some such on the existing foundation. That's as structural as it gets. Hire an engineer. I can;'t believe you can pull a permit without an engineered plan.
Lifting the building is not that difficult. I lived in the house that was moved 12 miles-similar sized to yours. Four I-beams were put under the house, it was jacked up, wheels placed under the beams, and it was driven the twelve miles overnight-set on the new foundation the next morning. Water, sewer and gas connected, back in business within 48 hours. In current dollars, with a new foundation-it was probably 40 grand in rural Colorado. Downtown Los Angeles might be pricier.
The points about the drywall cracking, and the doors not closing square are correct. Once the house is reset, the "twists" of the frame will be different-it will settle down in a year or so, but plan on some cracks and stuck doors. Nothing some spackle, paint, and a jack plane won't fix.