From Earth's point of view, any energy coming to us from the Sun stays in the Earth's system, land, sea, atmosphere as heat until it re-radiates into space.
The CO₂ in the atmosphere now trapping the heat will be here 1000 years.
Heat goes into the ocean, so maybe the Earth cools off.
No
The system, cup of tea/Earth can only cool when its heat leaves the system.
The CO₂ is a thermos keeping the Earth warm for 1000 years.
Stop adding more CO₂ & things will equilibrate somewhat warmer than they are now.