Adult playing a 3/4 cello?

You said that you got measured at a "Music Store". I would strongly recommend that you get a recount. Unless your arm length is very short for your height and your finger length is also short for your height, I don't see a need for a 3/4. I would go to a violin shop to get properly measured or talk to your teacher, they should have a 4/4 that they can measure you with. The other thing that most music stores don't know or carry, is a 7/8 size cello. If by some chance you can't play a 4/4, I am as sure as anyone that can't measure you properly, that a 7/8 would. Typically anyone above around 5' should be able to play a 4/4, so you should be able to.

That is too small. No doubt.

Taught strings in public schools grades 4 and up. I now work with a community symphony. We have smaller cellist all play 4/4 cellos (full size)

It isn't about the size...it is about the sound. If the sound is what you want...then who cares how large or small it is.

Nothing wrong in playing a 3/4 cello if that is better suited for you. Makes total sense.