So much of what you said about the lost belief of fans rings true to me. I can tell you all the games that made me believe that Texas was always going to win when a game was in a situation where it could go either way:
Football: 1998 Nebraska, 1998 A&M, 1999 Nebraska (in Austin), 2002 KState, 2003 KState, 2004 Okie St, 2004 Kansas, 2004 Michigan, 2005 Ohio St, 2005 Okie St, 2005 USC, 2006 Nebraska, 2007 Nebraska, 2007 Okie St, 2008 Oklahoma, 2008 Ohio St, and 2009 Nebraska. Hell even the A&M game in 2011 wasn't too far removed to carry that feeling out of that game.
I can add in a couple specific baseball and basketball games too: Baseball 2002 vs Rice in Omaha (the Quintanilla play from 3rd base) and 2005 vs Baylor in the Chance Wheeless/Kirk Gibson game. In basketball I think of the Kenton Paulino shot to beat West Virginia in 2006, Brandon Mouton hitting a 3 to beat Tech in 2004, and PJ Tucker hitting the buzzer beater against Providence in 2004.
The feeling you are talking about in the Rose Bowl in 2005 and in 2009, that feeling was earned through lots and lots and lots of winning. The feeling of anxiety lately, well that was also earned through lots and lots of losing. Thinking about the games that make me personally feel like Texas is cursed and living in a nightmare... 2011/2012 OU blowouts, 2014 UCLA (the coin flip), 2014 OU, 2015 Cal (PAT miss), 2015 Okie St (bobbled punt snap), 2015 Tech, 2016 Cal, 2016 Kansas, 2017 USC (OT turnover #1), 2017 OU, 2017 Okie St (OT turnover #2), and 2017 Tech (3rd and 2). For basketball this year alone you can add Duke, Okie St, and Tech.