more expensive solution, a saddle with longer rails or a seatpost with a longer setback. Basically if the bike is the right size for you (and you are not too far out of the mainstream when it comes to proportions), this can be fixed. A competent bike shop will set you up right. Where your center of gravity is on the bike matters a lot when it comes to stability.
It's pretty normal to move around on the saddle depending on what sort of riding you are doing, the terrain, wind, etc.
In any case, your description of the problem hasn't convinced me that you have a problem. It may be that you need more time in the saddle before you really know whether this is an issue or not.