I don't know anything about Thunderease, but have used Feliway for new cat introductions. It's not a miracle product but it definitely does help. When I brought Holly in, Maggie was not happy. I could definitely see a difference in her behavior when the diffuser ran out. I continued using the Feliway until it ran out and Maggie's behavior didn't change. Took about 4 months.
On a side note, the product I find works the best for calming in my dog is CBD. He gets it when he goes to the groomer. A few drops in their food might do the trick. It's expensive, but works way better than anything I've ever tried.
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Holly, Misty & their friend, Jake, the dog.
Onyx, Callie May, Maggie & Kobi forever in my heart.
Yes, I know I need a new signature pic...someday...