Can my soon-to-be ex-husband get full custody of our kids if I have been arrested for shoplifting and have once been on drugs?

The court will only look to protect the best interest of the child. Your criminal record, although hurting, does not prevent you from getting custody. You must present your best evidence why it is in the best interest of the child to live with you. You will have to find evidence to prove that the criminal records stay in the past and you are a better person now and more suitable to be a custodial parent.