If I receive a canonic annulation but I am still married by the law and I ask the State for a divorce, that would be right and that would be legitimal because when you marry by law they say you that you can divorce, but divorce can sound wrong for the people. How can I have a civil divorce without make people think that I agree with divorce?
IIUC, a Church annulment can only happen AFTER there has been a civil divorce.
That is correct in the US, but it might not be the case in other countries.
As for the OP’s question, I think the only thing you can do is make sure that people around you know about the annulment if it takes place.
It is the case in all countries. A civil divorce must take place before any Church annulment proceedings start.
The Church does not officially require this but it is the practical way of determining that the two parties have no realistic hope of reconciling which is something the tribunals are required to verify.
It is also the practical way to take care of many issues, from division of property to care of the children to making the division public. This way, the tribunal need not deal with those issues.
The requirement for getting a civil divorce first seems a little odd to me, because what if the marriage is found to be valid? Now you have a couple married in the eyes of the Church but not the State. How are they supposed to proceed from here?
Why would a still-married couple seek an annulment? The Tribunal will only proceed if there is no possibility of reconciliation. Not having sought a divorce means that reconciliation is always a possibility.
This situation happens all the time. It is why the issue exists of divorce and Communion.
But as others have said, divorce must come first so that the Church knows there will be no reconciliation.
That would have absolutely no effect on the civil divorce.
A Catholic may obtain a civil divorce for good cause without regard to the validity of the marriage.
What if many years later they decide, against all odds, they are falling back in love? Will the Church expect them to get married civilly again?
The Church expects them to live out their marriage vows in their separate lives. My parents did this after their divorce until they died 54 years later.