chinabar - if you group considerably better without a bow-mounted quiver, chances are high that you're tilting the bow when you shoot. Does your sight have a built-in level on it? If so, do you use it? If your bow is canting when you shoot you're going to be hitting either to the left or the right of your intended point of impact. If you don't use the same consistent and solid anchor every time - you're going to be shooting big wide groups, I'm talking groups that measure 6" or more at 20 or 30 yards.
Here's my ritual - for what it's worth...
Look at your target.
Decide on where you want to hit it (point of impact).
Keep your eyes on that little spot - stare a hole right into it.
Draw your bow from the bottom up, rather than from the top down.
Come to your rock solid anchor.
Put the sight window on the target - check the level - put the selected pin on the P.O.U. - check the level - check the P.O.I. - shoot.
All of the above takes approx. 15 seconds at a 3D shoot (holding on the target for about 3 or 4 of those seconds) an a lot less when you're hunting.