Successful though he was, Price's cartoon style strikes me as being more functional than distinctive. Fellow New Yorker cartoonists Charles Addams and Peter Arno, for example, had highly distinctive styles and became famous, unlike Price.
Examples of Price's work are below. Be aware that New Yorker cartoonists often (perhaps usually) did not come up with the ideas they illustrated: outsiders regularly submitted ideas to the editor.
Price did not work exclusively for The New Yorker.
Judging by the woman's dress, this was a late 1920s cartoon.
The girl is nicely posed and drawn. Clearly Price often dialed down his illustration skills for his cartoon work, but not here.
A cover from the earliest days of The New Yorker. I suppose the joke is that the man is eating, whereas the flappers are having coffee or tea, but I could easily be wrong.
New Yorker covers often made no special point.
Here, six or so weeks later, Price worked in a more solid style.
This cover notes the opening of the United Nations Secretariat building.