Pier and Beam foundation.
 

When our little dog walks through the living room the room shakes. I guess the piers may have settled during the drought. What do I need to do to stabilize the living room?

For starters, you need to put that dog on a diet. And stick with dog food, instead of that unhealthy people food you like to spoil him/her with.

But seriously, we don't know enough about the room's support structure to provide any meaningful suggestions. Maybe a more thorough description of your basement/crawl space details is in order, along with a few pix.

You need to inspect under the house and see what is going on. Could be under designed. Could be some of the piers are falling over. Could be rotten lumber or termites.

Send pictures from under the house. Also, check the baseboards. If the floor has sagged, a gap will often develop between the floor and baseboards. Measure the gap. These are good places to start.