When it comes to being married and living together,, there is no disputes of custody. Why can't we get our divorce first, then modify custody after?
Because a court wants to resolve all relevant issues at one time rather than stringing things out for a long time, but also because with a divorce comes finality of the marital relationship so the issues of child support, visitation, and custody get affected with the divorce.
Your question is unclear in reference to cheating and custody. Cheating has nothing to do with custody or visitation. It may effect alimony and distribution of marital property but parents typically don't get punished by losing custody because of cheating. When parties get a divorce, if there are minor children, the court will address child support, child custody, visitation & distribution of marital property all in the divorce. You are not able to simply get divorced and then address those issues later. And since you state that there is no dispute on custody, then there shouldn't be a problem.
Cheating may at best cast doubt on who should is better fit to have physical custody, but rarely does that. What it does effect is alimony. If you are divorcing your spouse because of adultery, and you have children, the divorce will deal with all of it. Answering the question of adultery will help determine alimony. Determining alimony may effect how the rest of the marital property is divided, which could effect support obligations, which could effect custody. I say that to say there is a method to the madness as it pertains to litigating all of these issues at once.
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