You’re looking for a dual controller, pretty common and many for sale here used. You need the voltage of the system and how many peak amps you plan to feed the motors which will tell you what controller you’ll need. Braking is done via the motors, but there’s no sort of mechanical parking brake feature. You’d need something mechanical for that.
I’m a traditional trigger system you’d flick forward for braking.
All four wheels rolling freely with no mechanical whine of sorts, yeah likely broken belts.