Heat while moving, none when I slow down or stop.

If the cooling system is functional, yes.
The ac condenser adds heat to the air coming thru the radiator when the ac is on. If the engine cooling system is weak, the extra heat will make the engine run hot.
Low coolant flow thru the radiator is a common cause.

Normal, the ac heat exchanger condenser need air moving across the fins to remove the heat. While moving in town the condenser isn't remove the meat that is building up the it's aluminum fins. Even though the vehicle uses an electric fan. The fan is used to cool the engine temperature. If you Toyota is equip-ed with automatic transmission. The electric fan will need to pull enough air to cool the engine temperature, automatic transmission fluid, air conditioner condenser, and maybe power steering condenser to cool the power steering fluid. Later model of vehicles/trucks are using different refrigerants instead R134A in which you Toyota uses.

The simples way to explain is the ac remove heat from inside your Toyota and this heat is removed by the refrigerants inside your ac in the car. To cool the refrigerant this coolant is removed to the heat exchanger condenser located in front of the engine radiator. So, while moving at 60mph there is enough air flow through the radiator, ac head condenser, and other aluminum heat exchangers. GB...stewbison

If it overheats on the freeway or constantly moving over 35 mph, it will more likley be the radiator blocked upped.

If it overheats only in slow moving traffic, it will likley be a thermo fan issue.

If it overheats all the time and is losing coolant it could be a blown head gasket.

turn the heater upto full heat (no a/c)to see if the temp goes down if it does then change thermostat.(i assume the fan(s)work)regarding the car accelerating on its own this is an anti stall feature on most cars(not all)especially VW group,it is also very comfortable in heavy traffic conditions when you dont need gas to move.(put the car in 2nd and crawl along with the rest)

Check your radiator fan is cutting in when it's supposed too Sounds like that the cause is no air flow when you stopped or in slow traffic

I had the same problem with my heater on an 05 altima. Was told it was the thermostat. Had it changed, problem still occurs.

you need to install a lower temp thermostat. and check the timing too far advanced will also cause this problem

this would be your cooling fan module located by your air filter mounted on the core support.

Are both doors closing tightly? Perhaps one is just loose enough for it to send signal that it's open. Even try sitting still with vehicle running and open one door at a time, and push lever of button that trips it, in with your thumb, to see if it stops. if you come across one that doesn't do it instantly, or it goes on /off with slightest movement, you probably found the culprit. Let's try the free/cheap/obvious things first.