Car will not idle after working on cooling system.

You still have an airlock in the system.

Many modern cooling systems are best filled using a vacuum filler - a new fangled tool that sucks all the air and remaining coolant from the cooling and heater system and then allows atmospheric pressure to force coolant into every corner of the system.

The traditional method of bleeding the cooling system is to disconnect the heater hose from the cold or return side where it connects to the engine (engine running at a fast idle, heater switched to hot) and be ready to reconnect as soon as coolant flows.
Other hoses sometimes also need to be bled and sometimes removing the temperature sensor can also help.

The air lock is primarily caused by the traditional thermostat jiggle pin being deleted though sometimes this function is replaced by the degassing system.
In the case of a really stubborn air lock or a recurring air lock it is worth checking for a degassing blockage.

Remove the cap from the surge tank. Set the heater control temp to max (A/C in any mode EXCEPT max). Start the engine and idle until lower radiator to pump hose is hot. Allow the engine to cool to ambient, and then check/fill the surge tank and replace cap.

Could be low on antifreeze. The heater core is the highest spot in the cooling system and it may develop an air pocket. Another possibility is the thermostat is stuck partially open and doesn't let the engine warm correctly, watch your temp gauge on the dash and see if it comes up to the middle like it should while idling. If it doesn't, it may need replacement.

The sloshing noise may be caused from air in the cooling system.

If there's air in the system, it ends up in the heater core because it's the highest point in the cooling system. When you accelerate, the water pump spins faster pushing coolant into the heater core forcing the air out causing the sloshing noise.

To remove any air from the cooling system, get the engine up to operating temperature. With the engine idling loosen the upper radiator hose clamp. take a small flat bladed screwdriver and slip it between the upper radiator hose and the radiator hose neck. Allow the engine to idle until all that comes out from the radiator hose is coolant. Retighten the hose clamp.

Sounds like the air blend door is not opening to let heat in.

Engine coolant is provided to the heater system by two 16 mm (5/8 inch inside diameter) heater hoses. With the engine idling at normal running temperature, set the heater-A/C controls as follows. Temperature control to full Heat, Mode control to Floor, Blower control to the highest speed setting. Using a test thermometer, check the air temperature coming from the center floor outlets and compare this reading to the Temperature Reference table.

If the floor outlet air temperature is insufficient, check that the cooling system is operating to specifications. Both heater hoses should be HOT to the touch (the coolant return hose should be slightly cooler than the supply hose). If the coolant return hose is much cooler than the supply hose, locate and repair the engine coolant flow obstruction in heater system.

POSSIBLE LOCATIONS OR CAUSE OF OBSTRUCTED COOLANT FLOW

• Pinched or kinked heater hoses.
• Improper heater hose routing.
• Plugged heater hoses or supply and return ports at cooling system connections.
• Plugged heater core.
• Air locked heater core.
• Restrictor in backwards

If coolant flow is verified and the heater floor outlet temperature is insufficient, a mechanical problem may exist.

POSSIBLE LOCATION OR CAUSE OF INSUFFICIENT HEAT

• Obstructed cowl air intake.
• Obstructed heater system outlets.
• Blend-air door not functioning properly.

TEMPERATURE CONTROL
If heater floor outlet temperature cannot be adjusted with the heater-A/C control temperature control lever, one of the following could require service:

• Blend-air door binding.
• Faulty blend-air door motor.
• Improper engine coolant temperature.
• Faulty heater-A/C control.

is there a chance that one of your heater hoses are clogged? or where they connect to the engine are clogged still? sounds like a clogged system. Cracked heater housing? Outside air bypassing the heater coil while driving?

I think that one still uses cables to control the blend doors and vent doors. Are you sure those are working and/or connected.

Also when you replaced the thermostat and heater core did you bleed the cooling system? With the engine cold remove the radiator cap and let engine idle up to operating temp while keeping the radiator topped off with coolant.

Are the hoses going to the heater core hot?

I’m happy to assist further over the phone at https://www.6ya.com/expert/jeremy_d728a59f986299fa

Just because the overflow reservior is full does not mean that the rest of the cooling system is also full. The engine itself is very low on coolant and running hot as indicated by the temperature guage showing hot and the cooling fans running continuously. You obviously have a leak in the cooling system caused by either a faulty hose or radiator which is leaking once the system pressure increases as the engine heats up.. The rubber hoses in the cooling system can go hard and start to leak where they attach to the metal piping around the engine as they get towards 10 years old. Thoroughly check all hoses including heater hoses and the smaller bypass hoses around the engine for splits or leaks. Stop engine and allow to cool, replace any defective hoses or hose clamps and then refill radiator. Restart engine with radiator cap off and idle engine with heater control on full until it reackes operating temperature while continuing to add coolant until the system is full once again. Allow the cooling fans to cycle on and off a few times to ensure there are no air locks in the system. .Hopefully you have not overheated the engine to the point where you have caused a problems with the head gasket which could allow coolant to leak into the sump. This will be indicated by the dipstick showing a milky emulsion instead of just oil on it. If the haad gasket has failed, it will necesitate removal of the haed, have it machined, and then replaced with a new gasket. Not a cheap exercise.

Impalas are known for this issue, where the heating won't work properly.... usually caused by trapped air/bubbles in cooling system or slightly low coolant level. Park car so that the right front side is elevated and remove the coolant OVERFLOW bottle cap (NEVER the radiator cap) while engine is running.. This should allow the trapped air to escape. Also add a bit more coolant if needed.