Hi passat,
Congratulations on your new mattress 
It sounds like your protector may be the wrong size for your mattress (possibly with pockets that are too deep to keep it tight on your mattress). I have also seen other similar complaints about the Tempurpedic protectors.
I would use one or the other but not both. The St Dormeir is water resistant but not waterproof (which for most people is fine) but is much more breathable than the Tempurpedic membrane type of protector you are using. It’s isn’t meant to add any comfort to a mattress although it will have a little more effect on the “feel” of your mattress than a thinner membrane type of protector. If you are more comfortable with a waterproof membrane type of protector instead of water resistant then I would probably look at one of the ones that are mentioned in some of the links in post #89 here that fit a little more tightly (less pocket depth) than the Tempurpedic protector.
Phoenix