I had no trouble managing my mug of coffee while driving stick. (Before cupholders were universal, and before texting).

It required some deft coordination to get out of the parking spot and getting up to speed, but I spent most of my time in fifth gear and staying there. One hand is just as free as it is in an automatic.

Texting while driving remains a bad idea, but I don't think a stick shift would prevent it. I could have managed to text just as well as I managed drinking coffee.