In the course of our search we found that there is a large apartment complex in the very heart of Livermore called Mill Springs Park that will accept our pets and it's within our budget and it's very nice. We are getting a two bedroom two bathroom place there. It has a small deck, a wood burning fireplace, central heat and air conditioning. It's also one of the newer complexes in the area having been built in 1990 and refurbished in 2000. Of course it has a swimming pool, hot tub, fitness room and club room as well. We will be signing our lease tomorrow morning and the movers will be here on March 23rd in the morning. Once we move in and get organized I'll take some pictures.
One of the best things about this place is the location. It is within a couple of blocks of the 1st Street section of Livermore, a traditional American style "main street" sort of place, lots of nice restaurants and shops there. It is a block from Safeway, about 3 blocks from Trader Joe's.
Here is the location:
Here is a video of the place: