What kind of drive "assist" system am I taking about?

I am looking to create a beach cart motor drive system that when it senses it's being pulled in a direction the motors will power the cart in that direction if the person pulling increases or decreases speed the motors will adjust accordingly if the person pulling stops the motors will stop. I would like the wheels to the only source of "input" (with the exception of a manual override.) I remember seeing a toy that was controlled in this fashion but can't remember where. I was wondering if there is a name for this type of control or a resource someone can recommend.

I would like the wheels to the only source of “input”

No.

It sounds to me like you need the handle to be the input. That can detect it is being pulled, something spring loaded with a linear displacement detector.