One thing too, when you're just working with stay, reward the STAY not the release. In other words, reward for staying but don't give the release command, just keep rewarding the stay, with cues etc. then release but don't reward the release, you want them to remember 'stay' as the command not release.
This way, as you move on to a stay while you're moving, the dog will have stay set in their minds. I did this with Ra Kismet and he will do stay in sit/drop/stand position and remain there even I fiddle with dog bowls in another room. It takes time to get to this point though, so keep working on the basics, as suggested above, while working on really setting the 'stay'.