This question about snare beds comes up from time to time. The snare beds on "acousti perfect" shells are very wide and shallow and gradual. They cannot be detected by the naked eye or running your finger along.
In order to check if it has snare beds you need to do a test like using a flashlight in a darkened room. Place the shell on a very flat surface and see if any light emerges in the area of the snare bed when you put the flashlight inside pointing out. The beds could be more than 1500 mm wide and only 1 mm deep. Have you done that sort of test yet? Or are you just inferring lack of snare beds because of the sound?
If you have the bottom (resonant) head tuned in such a way that it responds strongly to the batter head then the two heads will continue to excite one another after just one hit. In that case the snares are just doing their job, but what you can change is the relative tunings of the top and bottom heads. The answer may not be de-tuning the bottom head, or de-tuning the lugs on either side of the snares. Depending on how you have the top head tuned the answer may be to tighten it up.
Another thing to check is that your resonant head hasn't been overstretched at some point so that when you tighten it up the collar gets too low. I had this happen on my 2002 Black Beauty once when I was using a thin resonant (the thickness was 200 mil rather than the more common 300 mil from memory). The solution was to buy a new resonant head and don't over tighten it. I changed to a 300 mil one and haven't had the problem since.
With Ludwig anything is possible, but mixing up a Super Sensitive shell (with no snare beds) and then putting a standard strainer on it is quite unlikely. That doesn't mean it can't happen. But it is unlikely so it shouldn't be the first explanation you consider.
