Can I get residency in a state if I am renting?

That is a question to check with your states guidelines for residency. There has to be some proof that you live in the state at your rental address. Other bill history like cell phone, car insurance, work pay stubs?... Have you gotten a Florida drivers license or id? Check with your local government or start online to see what is required in your state.

Each state has their own requirements. You can start by obtaining a Florida Drivers License, vehicle registration. Visit Florida website for further requirements - https://www.stateofflorida.com/residency/.
You would probably need a copy of your lease, utilities to show that you are living in the state. Again, visit the website and your the local DMV for further requirements. I moved to Florida about 3 years ago and I needed a copy of my lease, utility, title/registration for vehicle, auto insurance to obtain Drivers License and establish my residency.