Stále is used for a continued action, and means that the actor doesn't end doing it. In this context it means that he seemingly never wakes up. A synonym is pořád.
Vždy is used for time. For example, "vždy můžeš začít". An emphasised version would be "vždycky".
A useful way to think about it is "vždy"="always" and "stále"="all the time".
Could this sentence mean either 'Her husband is always sleeping', or 'Her husband is still sleeping', depending on context?
Můj překlad: "Her husband still sleeps." mi systém opravil na: "Her husband still is asleep." "asleep" vidím prvně. Může mi prosím tento výraz někdo vysvětlit? Díky.