[ClumsyPilot]

Not at all, first our population will never exceed 11 billion, and most developed countries are in a demographic crisis. There is no sign of this trend reversing anywhere.

Second, what does 'earth is full' even mean? We have vast swaths of 'useless' land, like arctic and desert. Cities/towns/anything cover like 1% of the world. It is easier to desalinate water / build cities and greenhouses in deserts than it is to move people to another planet. We could host a lot more people if we adopted some serious geoengineering and built greenhouses /ate less meat.

Thirdly, travel to another star system requires insane amounts of energy, and could only be done by civilisations that already have enormous space infrastructure and industry. In which case you build habitats like we build skyscrapers, you can terraform, etc. In that case you don't need or want to ship billions of people to another star system.