If it is shooting perfect through paper, and you move the rest, it will no longer shoot perfect through paper. Typically!
Really, I think you are just trying to mask an issue with the switching to the feathers. If you cams are timed, your rest is "in tune", shafts are correctly spined, and you are shooting bullet holes in paper......then I would try a different method of tuning to re-affirm that you are actually in tune. Walk back is a good one. In short...hang a nail from a string on your target so that it intersects your "bull’s-eye", shoot at the BE at 10 yards using your first pin (20 yrd pin or whatever). Step back to 20 and repeat using the same pin. 30, and then 40, all the time using the same pin. Your arrows should be perfect left to right and jsut gaps from top to bottom. Like the buttons on a shirt.
Also check for cam lean. Cam lean is "OK" at either full draw or at rest, but not both. My bow is tuned very well and the cam leans to the left at rest. When I come to full draw it is straight as an arrow (PUN).....good luck!
Also.... a really good step you should take is to start at the basics. Check the axle to axle measurement and make sure it is correct. Measure the brace height ect. Just make sure it's at factory specs basically. Where are you located?