What would happen if a person declined oath to tell the truth in court? Said "I'll answer ?s but won't swear"?

No witness or juror is compelled to swear. I have been a witness and a juror a number of times and have never sworn an oath.

All courts in the U.S. allow for persons to affirm to tell the truth.

Usually, persons who believe that there is no justice in courts are persons who try to win without a lawyer. Not knowing the rules and the law usually means that they lose.

Persons telling the truth increase the likelihood that justice will be available.

A contempt charge is better than perjury? No, both will land you in jail.

If a person is voluntarily testifying, they will be dismissed. if they have been subpoenaed, they will be charged with contempt, the same as just refusing the subpoena totally.

It will be interesting to see how you high handed declaration that 'there is no justice' stands up once you've spent a day in jail.

"why should anyone feel obligation to tell truth if there will be no justice?"

Because if you don't, you are in fact contributing to the injustice.

It is not your job to judge the court. You obviously do not know enough about the law to do that. Your job is to show up and speak the truth. The only lawful way to avoid testifying is to invoke the Fifth Amendment.

It isn't actually necessary to swear. One can "affirm" which just says you'll tell the truth but leave religion out of it. However if you refuse entirely to promise to tell the truth, you will be dinged for contempt of court and no that isn't necessarily better than perjury. There's a guy who served 14 years for refusing to comply with the court's demands.

You would not be allowed to testify without an oath.

A witness who refuses to testify and does not have a testimonial privilege (i.e. 5th Amendment or spousal) may be held in contempt of court and fined or even jailed.