I read that for fixed stress (i.e. fixed elongation), I should observe decreasing strain over time.
This is a misprint. Fixed elongation is a fixed strain i.e. in this experiment you apply an initial strain and measure the decreasing stress as a function of time.
The phrase visco-elastic is a rather general term that covers a multitude of sins. However in general if you apply an instantaneous strain you expect the stress to start high then decrease with time as the material relaxes. The stress may decrease all the way to zero if the material behaves in a viscous fashion at long timescales (e.g. Silly Putty) or the stress may remain non-zero if the material behaves in an elastic fashion at long timescales (e.g. a rubber band).
I would guess you're thinking of non-Newtonian fluids, of which Silly Putty is a classic example. These tend to behave elatically at short timescales and like a fluid over long timescales. So when you first apply a strain you get a force given approximately by Hookes law, but over time the material flows and the force decreases in a roughly exponential way towards zero.
For non-Newtonian fluids if you apply a constant stress you get an immediate initial elastic response then the material starts to flow so the strain rate tends to a constant. Just to confuse matters these material are frequently also shear thinning, so usually the strain starts rising slowly then approaches a linear variation with time at long timescales.
Incidentally, the vast majority of lab rheometers are controlled stress rheometers that work in exactly the way you describe. They apply a predetermined constant torque to the sample and measure the resulting angular velocity.
