Unless I've missed any, I think this is a complete list of dual Final Four schools with the individual championship results per year:
2014 - UCONN (2-0)
2013 - Louisville (1-1, men's) (UCONN prevented Louisville from becoming the second team to win dual champs by defeating the Louisville women in the title game)
2011 - UCONN (1-1, men's)
2006 - LSU (0-2)
2005 - Mich. St. (0-2)
2004 - UCONN (2-0)
2003 - Texas (0-2) (UCONN women's team knocks Texas out of FF)
2002 - Oklahoma (0-2) (UCONN women insure no women's title by beating OK in champ game)
1999 - UCONN (2-0) (UCONN men insure no men's title by beating Duke in champ game)
Duke (0-2)
1983- Georgia (0-2)
Interesting that it happened five years in a row from 2002 to 2006 and nobody but UCONN ever one either title during that stretch. UCONN's overall championships record is 5-1. Louisville is 1-1 for their only season. Everybody else is 0-fer both titles every time. The overall championships record for all schools not named UCONN is a woeful 1-13. Unless you're named UCONN, it's been the kiss of death.
It will be a very long time, if ever, that the other schools will collectively improve the overall record to anything coming remotely close to UCONN. It would take a lot of 2-0 performances, and since that has only been done twice in the 34 year history of dual NCAA BB tournaments, it's not likely to happen any time soon.