This sounds like a wonderful opportunity, OP: for your career and your relationship.
I assume you're in fashion and he's in finance (or law). If so, you'll want easy access to Wall Street/Soho.
If you're on $10k after tax, without kids, you'll be fine. Once kids are in the picture, it's pretty difficult bc childcare is so expensive, but it's fine for you two. Once you get a job, it'll be even more.
You should probably be saving for a pension, etc., but you have time to settle first.
There's plenty of places to say. If he's on $190k, then you can rent up to $4750 on his salary alone, without anyone fussing at you.
I just checked StreetEasy for apartments in Cobble Hill, Brooklyn Heights, DUMBO, Tribeca, Williamsburg, SoHo, Tribeca, the Village, and Chelsea, at or below $4,500, and I found 302 apartments. (Some of them are really more Vinegar Hill than DUMBO and Bushwick than Williamsburg, but there's plenty there.) There are even things with pretty raw bricks that feel ridiculously Young Person in City (streeteasy.com/building/127-west-15th-street-new_york/4f)
Your bills will probably run about $1-1.5k per month. That then leaves you with about $4k to do with what you want: food, travel, drinks, etc. Not a huge amount of money in NYC, but not none.
You'll need a broker, probably, for the apartment. It's a pain the arse.
Enjoy your life!