What should I do if my house is leaning up off the foundation?
Being bolted down, or not, is an era-dependendent thing - modern houses generally are, many older ones are not.

Usually, the house has not lifted, despite appearing that way - what has happened is that the foundation has sunk - in this case, in a manner that moves the actual support of the house back to the points where the sill is still touching the foundation.

Mitigation ranges from the low-cost approach of shimming the gap to the high-cost one of a new foundation. If considering the latter, it may make sense (if you have space) to build a new foundation and move the house onto it, or to move the house off the foundation, build a new one, and move the house back onto it, rather than rebuilding it "in place." It's certainly posible to support the house where it is and build a new foundation under it, but it adds enough complexity to the job that paying to have the house moved once or twice is often the less expensive route. 

Beyond shimming this is not a great project to DIY, for the most part.