I've not used the adapters. To me it would seem like using high rings when you don't need to.
I think the Kwik Sites would be ok, though.
Your Savage may have Weaver Tip Offs designed for the 3/8" dovetails. Believe it or not, Weaver used to make a one piece mount for a British Enfield .303 that used the 3/8" dovetail rings. I've used them for years and they work just fine. You should still be able to find the Weaver rings that are not see-thru, if that's what they are.
Do you know how or why the scope rings moved? I've not experienced movement on bolt action before, unless there is a problem with the dovetail. Semi-auto's are known to occasionally cause the scope to inch forward due to the slamming of the bolt in the receiver. Perhaps the dovetail is a bit damaged or rounded? Is there enough room left to relocate the rings? you may have to reposition them on the scope tube.
What I'm getting at is, if the dovetails are buggered, you'll probably have a difficult time with any mount.
If they are, it may not be the end. It's possible to re-groove a dovetail with a sharp file enough to get a decent bite.