Universe is huge. Wars on earth are waged over scares resources. If we are capable of reaching aliens would we really be incentivized to fight them while there are so many resources available elsewhere?
There is a mismatch here that you are implicitly claiming that is solved for every space-faring civilization. Many people explicitly make this claim, what is reasonable, but it's not good to keep it implicit.
In the early years, sure, no problem. But conceivably the pace of resource utilization will increase as well. Eventually they will conflict. Worse, greedy civilizations would probably be selected for.
And how long before random differences and exponential growth mean the other side gets an overwhelming power advantage? Historically in human societies, that is essentially never a good thing.
Be careful not to conflate public justifications with actual motivations.
(I don't think Iraq was actually such a situation, but at least on the American side some influential decision makers may have believed that it was.)
On the other hand, you might have something more like a cold war situation, where both sides have the capacity to destroy the other completely or almost completely but not without an equivalent counterattack, and so both sides have much more to lose than to gain by striking first.
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.
Are they? That was once a common belief, but recent results in extrasolar planet searching would tend to contradict it, or at least cast it into serious doubt.
> Once the earth is full we’ll have to find a way to move
Will we? It's quite possible we'll see humanity's maximum population within the next century. Malthusianism didn't really survive contact with modernity; it turns out that most people don't particularly _want_ to have fifteen children, and as countries develop their population tends to become self-limiting. Wholesale emigration off earth feels like a very unlikely solution to population pressure, especially given that society seems to be automatically solving it.
And if we have the energy to lift billions of people off earth, we also have the energy to massively increase population density. Food, in particular, is ultimately largely a question of energy; we typically grow it in fields today, but given super-cheap energy there are other options.
