IMO there is no connection between preferred orientation and peak shift. The angular position of a diffraction peak is related to an interplanar distance beteen crystallographic planes in crystallites (Bragg equation). The preferred orientation affects mainly the intensity of the reflection, since there is more/less crystallites with certain orientation.  If you have recorded the peak shift it is rather connected to the mechanical strains than to texture. 

The preferred orientation  such as the wire texture or the  sheet texture only affects  the  intensity and  line width of XRD peak but not the peal position, which is closely related to the interplanar d-spacing of that family of planes responsible for the x-ray diffraction.

As the colleagues argued above, the orientation had nothing to do with peak shifting. Furthermore, different sources of Ka provoke different positions of the same profile, e.g. the main signal of anatase is located at 2theta = 250 (KaCu) and at 300 (KaCo). A change in interplanar distance d can be provided by doping - to put it differently, by solid solution formation. As long as you study the latter phenomenon, you'd better analyze the peaks found at  2theta > 500. This is because error decreases with increasing theta (see an illustration attached). The error of d can be estimated by derivation of the Bragg equation.

Preferred orientation in Powder XRD didn’t affect the peak position but the intensity of the particular plane i.e preferred plane and its parallel planes (example 111 & 222) has the high intensity compared to the standard values (which is closely related to the interplanar d-spacing of that family of planes as per Prof. Tarik’s explanation). X-ray texture measurements will give you more details. Moreover, preferred orientation effects can be possible in electrodeposited films, oriented fibers and rolled sheets.

I can imagine one case when the apparent peak position is affected by preferred orientation. If there is a low-symmetry system with strongly overlapping Bragg peaks (i.e. several Bragg reflections contribute to the observed peaks, event at low 2theta angle - this is the case for large unit cells, common for the organic compounds) the alteration of the intensities can be seen as a slight shift of the peak position.

I have never observed this, but in my opinion, it is possible. Just a special nasty sample is needed.

We should remember that the cubic, sheet or wire textures in metals and alloys are obtained by the severe plastic deformation such as extrusion and or deep drawing operations, which create large amounts  of  not only dislocations but also the deformation stacking faults and twins. Where especially the deformation stacking faults -as ı have mentioned at this platform previously-  causes appreciable amount of peak shifts in (111) and (200) lines in opposite directions in FCC structures, such as that  (111) shifts  towards the larger angles. That means if one  makes an XRD measurement on the as-received  sample having textures without exposed to any annealing treatment (recrystallization) previously,  then it is sure  that  he/she  may observe peak displacements compare to the XDR  line spectrum of single crystal unstrained samples.                                                      .