Praise be to Allah.
If the urine stops after this time, then you have to wait, and there is no concession for you in prayer when there is najaasah (impurity) present, because you are not regarded as suffering from incontinence. You should choose times when there is less crowding so that you will not disturb people with your being late.
As for putting a piece of cloth over your private part, that is not something that is forbidden when in ihraam.
We asked Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him): What is the ruling on the muhrim (pilgrim in ihraam) who suffers from incontinence putting something over his private part, such as a bag, so that the najaasah will not spread? Is this one of the things that are forbidden when in ihraam?
He (may Allaah have mercy on him) replied: That is not one of the things that are forbidden when in ihraam.
And Allaah knows best.