I definitely feel your pain; as I am a native NY'er whose first design opportunities were in major design meccas (NYC, Seattle and Chicago); both corporate and consultancy. After graduating in 2011, the competition was so fierce in NYC (even after meeting with design directors and getting great feedback on my portfolio) that I had to reevaluate my dream of ending up at an award-winning consultancy and start casting a wider net in search of ID work elsewhere (in not-so metropolitan cities) if I were to be gainfully employed.
I finally ended up being interviewed and hired by a major company in the Midwest known for highly valuing design and their employees; and I haven't looked back.
I don't miss the big city and have to come to embrace small town values; and I am close to some major cities if I feel the need to venture out. I am compensated quite nicely (starting salary 60K; and in three short years, am closer to 70K). Granted, the projects aren't as diverse as a consultancy, but that's ok. We have multiple product categories, so it's possible, even encouraged, to move around. I pay only 600 dollars a month for a nice one-bedroom apartment in a nice, modern complex. And in 3 years, I've been able to put a huge dent in my student loan debt. And I don't have to eat Ramen every night. I love the work I am doing, have a great boss and great team members. And I don't feel exploited nor overworked.
When you wrote, "I feel like my boss has the attitude that “there are a million other recent grads who are begging to have my job, and I’m lucky to be employed at all.”... I can sympathize because after talking to established designers over the years and reading gripes on Glassdoor.com about said consultancies, you will see that their name on your resume is your true compensation (not to mention the great work most of them do) because for the most part, designers complained about feeling exploited, overworked and under-compensated. No thank you!! Basically, you have to truly enjoy the work that frogs, Smarties and their ilk do because you won't get rich doing it. I made an assumption you are working for a design firm; either large or small, so I apologize if I am incorrect. (My advice below still stands)
You will have to ultimately decide if being in a large expensive city is worth the financial heartache; or if you just need to move onto a company/consultancy in your same area that will compensate you what you are worth. The problem is that in major design meccas with a oversaturation of designers, the competition is not only fierce, but employers feel (for the most part) encouraged to underpay their entry level designers because they know that there will always be another young talent around the corner. Design is a fun, rewarding business but can also be cutthroat and ugly if you allow it to be. My suggestion to you is to either go corporate and/or move to a city with a lower cost of living; and climb up the corporate ladder. When you are then financially more sound (debts paid off, money saved up) and more established as a designer, you can then venture out to the big-name consultancies or to the design meccas you dream of and demand the compensation you deserve; if that is your wish.
Best of luck to you!