she is trying to be fair as I am assuming you paid for the last 2 meals. She just wants to hang out with you, and thinks dinner out is the way to go, so she is willing to pay for it. Option 1: go out for dinner, you pay, but make sure you are polite about it (you always pay)
Option 2: you say thank you for the offer, and instead of going out, you suggest her coming to your house, or you going to her house and you guys cook a meal together, try something new to both of you, out of your comfort zone. Finish off the night with either a movie, or a walk somewhere kind of romantic, like by a river or some shit like that...
BUY HER A SMALL FLOWERED PLANT, nothing big, say like $10-20 little counter type plant.
DO NOT GET DRUNK, I repeat, do not get drunk.
She sounds like a keeper bud.