What is the average cost of raising a sinking foundation and correcting the sediment beneath? ~more or less~?

It's quite expensive as it's a great deal of work to get it fixed. You should research three good, reputable foundation repair companies in your area. Have each one come and give you a price for the repair plus exactly what they will do. Then pick the one using the method and cost you feel most comfortable with. Leaving it is a bad idea as it gets worse quickly and eventually internal wall, window and door damage adds to the cost.