re: Could a new thermostat fix this issue I’m having with nighttime cooling?

Thermostats have an temperature off-set, usually adjustable, of a couple degrees or so on either side of the set point, before the unit turns off or on. It prevents rapid off-on cycling of the unit. Look up your thermostat model number see if there is an operational manual and read he manual to see if it is adjustable.

Is this a digital or analog thernostat?

Might be time for an thermostat upgrade?