Too murky to know for sure. Driving with a hot motor isn't good, so don't do it. Very basic: you burn gas to get power, even a little bit of power. The gas burns in the combustion chamber. The combustion chamber is part cylinder and part head, that's why these components have coolant passages in them. When coolant doesn't flow sufficiently, the coolant turns to steam and pressurizes the whole system beyond what the radiator cap allows. (This is why you shouldn't open the radiator cap on a hot motor- by releasing pressure, you can make a lot of steam, which makes coolat spurt out the radiator and will burn you. Don't do this, either.) Back to an overheated motor- when some of the coolant turns to steam. The steam pushes coolant somewhere- out the radiator, maybe it bursts a hose, or maybe squeezes through the head gasket. None of these is good. If you keep driving, you lose more coolant, until the motor gets really hot, and then very, very bad (read "expensive") things happen- you can fuse the pistons to the cylinder, you can burn a valve, you can warp the head, or you can melt a hole in the top of the piston.
Before you drive this car again, wait till it cools. Then fill the radiator and note how much coolant it takes. If you put in a quart or so, you'll probably be OK but only to take it to a good shop and diagnose the problem.
The problem? Could be the radiator or the thermostat (seldom the thermostat, but worth checking) or something easier- maybe the belt that drives the water pump is slipping. In any case, 1) make sure the radiator is full before you start the motor and 2) don't drive if the temperature shows it's hot. And don't put off diagnosing and resolving this. If you ruin the motor, it'll be expensive to replace. ... Best of luck!!
PS- Automotive Maintenance 101: Do not replace something just to be "doing something." Make sure you need a new radiator before you buy one. Ditto the water pump and drive belt. And any other component.