Is my water pump going out?

It could be a few different things but you could start with a thermostat. Either way, definitely take your car in for a checkup the worst thing you can do to an engine, besides running it with no oil, is to run it hot. Once it starts to overheat a little, that is your warning that it is on the way towards overheating a lot. I have seen people who are "too busy" to have a thermostat replaced '($25 part or less and minimal labor) and end up seeing them a month or 2 later with a blown head gasket. Take care of the little problems and you won't have big ones

Just to add it could also be as simple as a coolant temperature sensor (also inexpensive) causing your cooling fan not to turn on. This would explain only heating up when not moving.

Thanks, I'll check into those 2 things, then have my dealer run a diagnostic on it.