My guess is that the plastic cases with which you are familiar are most likely some of the cheapo Chinese knockoff ones which very often do have that slick surface (which you term as slimy)
But Polyurethane comes in a variety of types, textures and thicknesses.
There are two brands which make plastic cases which have a soft velvet-like texture to them. Both the ifrogz Luxe line as well as the Incipio Feather styles have a very pleasant feel to them.
Put each line into search along with ipad on Amazon and you’ll see full color pics and it’s obvious that they don’t have that shiny slimy look to them.
The reason I know how nice they feel is because I’m currently using my ifrogz Luxe blue one and I alternate it with my Incipio Feather one in Violet. Once you feel the surface of either of these models, I’m sure you’ll like it.
I fell in love with the feel the very first time I touched one. I currently like my blue one because it’s a metallic blue which looks really cool.
The only reason I suggested looking at Amazon is because they’re less expensive there, depending on the vendor; but, you can go to each company’s website also. And you can check out local stores in your area as well. But you’ll likely pay top dollar for the exact same product.
Both of these brands have an excellent longstanding reputation for high quality extremely solid and well made cases for Apple products. Their products usually get favorable reviews and high marks from tech review sites.
(altho I was initially a bit put off by Incipio because the first iPhone case I ever bought was an older model of theirs with really soft Polyurethane which was very stretchy and picked up lint and cat hairs like crazy. And it was black to make matters even worse. Could never get it clean. Obviously I returned it to ATT and picked up a Belkin Wave. But I don’t even think they make that style anymore,)
Their Feather line is totally different A very hard plastic with a soft feel to it’s surface. Hard to describe but sort of like Velvet.
And the Feather is also one of the thinnest, lightweight, but toughest cases put there. When it first came put for iPhone, everybody was raving about it all over the place. There’s even a neat vid on YpuTube in which the head of the company describes their manufacturing process and how they manage to make it so thin. Interesting. Great company.
So check out both of these companies and you’ll find something you like.