What is the appropriate sentence to describe something that someone was used to do at the early stage of coming to the USA? I would say:
"When I came to the US and want to ask Walmart's employees about something, they would ..."
I believe the problem of When I came to the US is that describes the moment of entering the USA and not the early stage (e.g., the first few months).
When I first came to the US ...
makes it clear that you are talking about something that is no longer the case and that something started the first time that you entered the US rather than on a subsequent entry. You are correct, however, that it leaves entirely unclear whether that something lasted for the first second, the first day, the first year, or the first decade. You can clarify that by explicitly mentioning the duration
In the first few months after I first came to the U.S. ...
tells us that the period is ended, that it lasted a few months, and that it started the first time you entered the US rather than on a subsequent entry.
There are lots of words in English. You are free to use all you need to make sure that you express your intent clearly. It may be good style to be succinct, but it is not good style to be succinct at the expense of expressing your thought fully and clearly.
You can phrase the sentence like this :
When I was new in the USA and wanted to ask the Walmart's employees....
or,
In the early months of my staying in the USA, I wanted to ask....
N.B. it will be wanted as it is in the past.
