Congrats on the new addition! One of my bassets whines a lot when he needs to go potty and when I leave the house. The other does not; both are rescue pups. I've noticed that since I added a second basset, my whiny one does much better. He came from a home with a lot more dogs, so it may take him some time to adjust to a new household.
A thunder shirt may work as well. My older basset, Copper, was advised by our trainer to use a Thundershirt to reduce his anxiety. It seems to work well. If anything, its a good layer of warmth for him.
Rescue Mom to Copper 52 pounds (DOB 7/10--Adopted 10/11) and Apollo (pitbull) 16 pounds (DOB: Approx. 9/22/12, Rescued 11/3/12)
Former Foster Mom to Jamie doxie/basset (A 6/12), Taz & Tommy (trans. to med care 4/12) Ashley (A 3/12) Gwendolyn (A 2/12)and Dudley (A 8/11)