Folk, a former Pro Bowl selection with Dallas, converted 19 of 25 field-goal attempts last season, but three of those misses were from 50 yards or more.
Terms of the deal were not immediately disclosed.
He started off strong last year by making his first 11 field-goal attempts, breaking the team record of 10 straight from the start of a season, set by Pat Leahy in 1986 and tied by Jay Feely in 2009.
Alexander was signed late last season after he was waived by Miami, and played in one game but was inactive the last two games of the season.
Campbell, an undrafted free agent out of Temple, spent part of last season on New York's practice squad.