Could shielding (not grounded) affect signal integrity?
You asked if shielding affects signal integrity: Yes, of course it does. Shielding changes the characteristic impedance of the transmission line model. 

Concerning your ungrounded shield: It might reduce external magnetic field coupling (inductive coupling) a little, but it does certainly not reduce magnetic field coupling between the cable signal lines. And an ungrounded shield does not prevent electric field coupling (capacitive coupling), too.  

So you might want to ground the shield. And you might also want to follow Mattman944's advice to use an adjacent ground return for each signal to minimize loop area.