Increasing the shear rate will reduce viscosity towards base viscosity but at higher shear rate, the viscosity seems to slightly increase as if shear thickening.
It is common. However make sure that the "shear thickening" is not due to secondary flow (sometimes incorrectly called turbulence). If the sample is low viscosity, the flow lines can change at high shear rates causing extra viscous dissipation, which looks like shear thickening. Try Googling "Taylor number flow" if you need more information.
Chai, "No" is the short answer. You might try changing geometries (plate-plate, cone and plate, Couette), but I have never compared their limits of pure shear flow, so I cannot advise you which is better. Sorry.