Any movement of a scope is worth worrying about....the longer the shot, the more amplified that movement will be.
From what you are saying it sounds more like the bases are loose as opposed to the rings. I would take everything off and start from the bottom up...re-tighten the bases down and use either blue loc-tite or clear fingernail polish on the screws and ensure they are solid and don't move. Then remount the rings doing the same thing (using loc-tite or nail polish on those screw threads as well). Pay close attention to the screws on the bases (especially the forward bases) and make certain they do not bottom out and make contact with the bolt (not uncommon on the Savage 116)...if they do bottom out, locate slightly shorter screws.
Also, as is common with the Savage rifles...make certain you have the correct bases...the older non-accu-trigger versions use a rear "flat top" action that isn't the same as the newer accu-trigger action that has a rounded rear receiver area where the bases are attached.