She has plenty of company among gay and lesbian couples in other conservative states, although thus far only a few have pursued divorce cases in the courts.
Even as the number of states legalizing same-sex marriage will soon grow to 16, most states like Mississippi refuse to recognize such unions or to help dissolve them. Gay couples who move to those states after marrying elsewhere face roadblocks if they wish to divorce, as do couples from those states who make a brief foray out-of-state to get married.
Often, such couples in non-recognition states would have to move back to the state where they were married and establish residency in order to get divorced an option that can be unworkable in many cases.
"The idea you can't go to your local courthouse and file for divorce is very disruptive," said Peter Zupcofska, a Boston lawyer who has represented many gay and lesbian clients in marriage and divorce cases. "It's an enormous waste of effort and time."
The right to divorce isn't as upbeat a topic as the right to marry, but gay-rights lawyers and activists say it's equally important.
"The marriage system is a way we recognize and protect the commitments people make to their partner," said James Esseks, director of the Lesbian, Gay, Bisexual and Transgender Project at the American Civil Liberties Union.
"Part of that system is creating a predictable, regularized way of dealing with the reality that relationships sometimes end," he said. "Those are the times people are the worst to each other, and that's why we have divorce courts. There's got to be an adult in the room."
On a recent evening, in the one-story brick house she shares with her two children, a new girlfriend and several pets, Czekala-Chatham sat on the edge of a leather recliner, shaking her head.
"Why should I be treated differently, you know?" she said. "When the courthouse is a few blocks from here, I should be able to walk up there and get married. I should also be able to go up there and get divorced."
She could get a divorce in California, but her lawyer argues that Mississippi wouldn't recognize the divorce and their marital property would remain "in limbo."
Melancon, who now lives in Arkansas, declined to be interviewed. She said in an email that she wants the divorce, but the "avenues to pursue are vague and expensive." She did not elaborate.
The Mississippi Attorney General's office filed a motion to intervene on Nov. 15 that said the divorce petition should be dismissed.
Mississippi "has no obligation to give effect to California laws that are contrary to Mississippi's expressly stated public policy," the motion argues. "That legitimate policy choice precludes recognition of other States' same sex marriages for any reason, including granting a divorce."
Legal experts say getting Mississippi to recognize the marriage for any purpose is a longshot. Lawmakers amended state law in 1997 to say any same-sex marriage "is prohibited and null and void from the beginning. Any marriage between persons of the same gender that is valid in another jurisdiction does not constitute a legal or valid marriage in Mississippi."
In 2004, 86 per cent of Mississippi voters approved an amendment placing a ban on same-sex marriage in the state constitution.
In his arguments for a divorce, Czekala-Chatham's lawyer, Wesley Hisaw, cites a recent ruling by the US Supreme Court that struck down parts of the federal defence of Marriage Act and ordered the US government to recognize legal same-sex marriages. That has created a situation where same-sex couples "are married lawfully under the laws of the United States, but not under Mississippi law," Hisaw contends.
He also argues that bigamous and incestuous marriages are considered "void" in Mississippi, just like same-sex marriages, but bigamy and incest are also grounds for divorce.
"There can be no legitimate state purpose in allowing bigamous or incestuous couples to divorce and not allowing the same remedy to same-sex couples," he wrote.
At least one same-sex couple has been able to get a divorce in a state that doesn't officially recognize same-sex unions. In 2011, the Wyoming Supreme Court ruled that two women married in Canada could get a divorce in the state, reversing a ruling by a district judge.
-AP