The two states say that abortions are nonessential and thus are required to be delayed in the midst of the coronavirus pandemic.
The actress refuses to work in the state after it passed the fetal abortion law.
Democratic Gov. John Bel Edwards has signaled that he would break party stance and sign the bill into law if it crosses his desk.
"This isn’t a scene from 'The Handmaid’s Tale.' This is happening in Alabama ― in our country ― in the year 2019,” she said in an email to supporters.
Missouri's eight-week ban on abortions is not expected to take effect unless Roe v Wade is overturned, with legislators saying it is not meant to be a challenge.
Democrats in the Alabama Senate attempted to reintroduce an amendment that would make exceptions for victims of rape and incest, but that motion failed.
The new law will ban abortions once doctors can detect a fetal heartbeat, which usually occurs at about six weeks of pregnancy before most women know they're pregnant.