Eating some beef broth

And there was a little bit of a carrot in there, broken up, and I swallowed it without realizing.  I've removed the pieces of carrot now, but should I be worried about that piece of carrot causing a dumping issue?

No.

Brevity is the soul of wit.  Thank you.  I did not have any ill effects.

Kiddo, you will more than likely not dump from a small piece of carrot. It might not sit well with some people that are early out. Please review what dumping syndrome symptoms are. There is a bevy of info to be found via Google or other search engines. Welcome to the other side!