I am not happy in New Jersey. I am debating moving.?

[Kevy]

DC has a higher cost of living. Yes, they have higher wages too generally, but that may not necessarily be the case in your line of work.

You should market yourself on linkedin and actively search for jobs before cutting ties with your current employer. Some jobs will pay for you to travel down on amtrak and interview, if they are big deal enough and if you have an advanced degree. However, if you cannot afford to travel there, a phone interview is appropriate. Remember, the bus goes to DC too, but if you take a bus to an interview, you really must allow extra time. I would recommend you take amtrak down to be fresh and on time for an interview, then take the bus back when you can be more leisurely about it.

"just get a job in Washington DC, quit the state job and move to Washington DC once I get a job there. Any suggestions?" This sentence is way out of order. I hope this is what you meant:
1. Get a job in Washington
2. Quit the state job
3. Move to DC

omit the "once I get a job there" part because you already said #1. Do not do #2 or #3 until you have #1! Very important. I'm not kidding, rent in DC is probably 50% higher than you're used to paying now, unless you live in extreme North Jersey.

When quitting your state job, consider the effects this may have on your pension and retirement. Should you stay 10 years minimum in order to get "vested" even though you don't enjoy the work you went to college for? Depends on how much "hard work" and time you have already invested. But who is to say you will enjoy your job in DC more?