River is NOT the character for you if you want a monogamous LI. River is unabashedly slutty, and she's not going to change.
Focus on either Techie or Amazon if you want a girl who won't be interested in anyone but the MC.
Edit:
DelinquentProductions, I'd love there to be a "I'll save you, River, from this life of debauchery!" white knight option, and just have River burst out laughing. "Oh, sweetheart...I am loving every single minute of this! Now if you'll excuse me, the Captain's asshole has a date with my tongue! Toodles!
"