I have just started a new a job and they do not have anyone who can answer my question, they claim that they cannot legally give me advice. So my situation is kind of weird. I currently live in DC and just graduated college and am living here the remainder of the summer. My parents live in Florida and still claim me as a dependent so I can claim Florida as my residency and avoid state level income tax. The reason why I am confused is because no one can tell me where to file my state W-4. So the situation is I currently live in DC, I work in Virginia, but Florida is where I would like to claim residency so that I can avoid the state level income tax. In which state do I need to file a state W4. If I file in Virginia because that is the state I work in will I be able to get the state taxes back at some point. If someone could help I would really appreciate it.

It does not matter what is your tax home. If you work in a state other than you tax home, you must file the tax return and report income earned in that state.
You may like to read this: Your U.S. Tax Return: Working in two or more states

You can probably get away with NOT paying DC income taxes for the rest of the summer, but if you are living there come September, you have a legal obligation to pay DC taxes.

You pay VA taxes regardless of where you live, because that is where you are earning your income. You need to submit the VA-equivalent form to the federal W-4 to have VA taxes withheld from your salary.