Although you don't provide a complete description of the facts, I'll assume that either trial is still pending on the DUI's, or you've already plead or gone to trial. That probably means that there are either substantial restrictions on your driver'ds license, or it's been suspended completely. Either way, that would substantially affect your ability to transport the children, either for regular events and exchanges, or in an emergency.
Also, if the DUI's occurred AFTER the agreement to joint custdoy, your ex has a serious argument, which it sounds like the judge may have accepted, that TWO DUI's show your judgment and decision-making is bad enough that there is a reason for the court to limit your custodial time.
You'll need to convince a judge, at some point, that you've taken serious and consistent steps to address the problem.
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