How does your girlfriend feel about those options?
Truthfully, in my book, honestyu really is always the best policy. And I tend to feel like your grilfriend does about dating people who aren't out. It's terribly hard.
And you CAN move out and be on your own at 18. Truth be told, until very recent times, very, VERY few folks at that age, save royalty, would not be expected to be autonomous.
So, I say be honest and be sincere. Explain to them that this is your girlfriend, that you love and care for her, and reiterate that so far as your heart tells you, you're queer and this is whom you love. Make clear that while you're on their dime, you're open to negotiation and accept that they do make the rules but that while your views differ, you'd like to be respected for who you are and would like to work towards that with them, especially since it hurts like hell. Offer to find a PFLAG meeting.
Might be worth letting them know, as kindly as possible, that even if they disallow you to date those of the same-sex while you're living there, which you'd accept but be very unhappy with, you will be doing so the minute you leave, and that everything we know about orientation tells us that just like you can't "turn them" gay, no one else can make you straight, and those attempts feel very dehumanizing and hurtful. if you have a family member in your extended family who can help, call on them.
I know it's hard to be sensitive, but everything we know about bigotry and bias also tells us that a softer approach is always better: it's stuff that is very, very hard to unlearn, especially for older adults, and likely they feel the way they do because those older than them taught them as much.
Editor and Founder, Scarleteen
My epitaph should read: "She worked herself into this ground."
-- Kay Bailey Hutchinson