Do your down spouts go into drains located next to the foundation?
How old is the house?
What type of foundation do you have, poured concrete, block, slab, fieldstone ?
1 thing I can think of is drilling test holes around the foundation to a depth of 2' below the footers and see if they fill with water and how high it gets.
The kitty litter smell is common when carpeting gets wet from moisture and / or flooding. The house moving and the kitty litter odor may or may not be related.