Hi - chartered structural engineer here. Definitely dig your own trial pit, take dimensions of the foundations - width from the wall face, depth, note the founding strata (clay, made ground etc), take photos. This is the first thing I would ask for.
But if you have cracking already, sounds like you already have problems. Could underpin with mass concrete but time consuming, do mini piles or screw piles, loads of options or potentially just re-build might be easier to get the extension you want.