If it's the same type shirt I'm thinking about, it was one of the original type t-shirts that Zazzle offered way back when.
It was the Hanes Soflink t-shirt that was specially made by Hanes for companies that printing using dye sublimation. And yes, they did feel weird (at least to me).
I still have one of those old shirts that were done via that method.
The good news is that Zazzle has a huge selection of t-shirt and garment types now. Most of them are the normal, popular brands like Alternative Apparel, Hanes (not softlink), American Apparel, etc. I don't know if they still do dye sublimation, but I think their main type of t-shirt printing has moved on to Direct to Garment (DTG) printing, which doesn't require specific t-shirt blends.
So, short answer to your question, you wouldn't have to worry about that weird feeling anymore
Actually, I just looked on their site and it seems they still carry that particular shirt as their "Premium" shirt with "premium" printing:
