We’ll just have to see. I don’t think we fully know the intentions of the Skrulls yet specifically enough to determine why didn’t they.

The Skrulls doesn't just want a place to live, They wanted their own planet which both Carol and Fury promise them.

Earth

Here in my hometown I have some places they can stay...

I think the Skrulls wanted a world to call their own.

I believe that was the main purpose in Captain Marvel which Carol was planning to help them look a home for them

Now my only guess is that Gravik has his reasons for wanting to take over Earth

Why don't they move to New Asgard well seems like I think that place is just for Asgardians and not sure if Skrulls would accept them into their home

Some potential answers

- Skrulls like to live in places with radiation, so it's unlikely that they would prefer that

- unlike the Asgardians, they don't naturally have a human appearance, potentially leading to a lack of acceptance by humans. We've seen what humans have done to Inhumans (and those were still have human), and we've seen what the mutants may have to deal with.

-Additionally they would probably have to always be shapeshifting, meaning they couldn't be themselves.

- And just like in the real world, Skrulls can both be refugees and prejudicial against humans, and even Asgardians.