I have these same issues (some of my teams are different), but I decided it doesn't make any difference - the sports are unrelated so you don't have to decide.
My conflicts are when it's competing teams in the same sport. I'm an Oklahoma alum. I've had a couple major opportunities to have to "decide". One was the 2002 Women's NC game. It was clear to me way before the game that for WCBB I was a UCONN fan. UCONN had played OU before, so I had already crossed that bridge (and when I was at OU they didn't even have a WBB team yet, plus as much as I loved Sherri, Stacey Dales was such a bih****).
The other was a UCONN-OU men's game in Storrs early in the 2003-04 season when UCONN was ranked #1 and OU was #3. Went to the game with an OU alum/CT resident friend. Neither one of us knew who to root for, so we decided just to go with whatever our hearts told us in the moment. We wore OU shirts under our UCONN shirts so we could take off the UCONN shirts if that's how our hearts told us to root. It took about 3 seconds to decide (both of us) - UCONN all the way. Gampel was 10X louder than any women's game, and UCONN blew them out from the beginning. No need to change shirts.
BTW this doesn't apply to football. Of course I would sell my soul, and my Yankees, etc, etc, for OU's 8th football NC.