If a person entered twice being caught at the border one time, would this new waiver work to gain residency if they are married to American?

There is no new waiver.  The regulation that will take effect March 4th merely allows someone already qualified for the unlawful presence waiver to apply for that waiver from here within the US.  You didn't indicate whether the person in question is in the US or not.  If they are, they could use the new process to apply for a provisional waiver from here in this country.  That person would still have to leave to process the waiver at a US consulate abroad.   The person would have to show extreme hardship to a US citizen spouse or child to qualify for the waiver.  As long as the border crossing attempts ending the person merely being turned back (and not arrested and excluded/removed) those incidents shouldn't affect the waiver process.