Should I only use my credit card?

I currently have a $5500 debt with only $6000 available credit on my credit card after a rough patch. My credit is just getting back to a 620 so I don’t qualify for a balance transfer card anywhere currently. I make about $4800 a month now, but live in a high col area so I have around 3200 in total monthly expenses.

Should I move my money into my credit card account from my checking account every pay period until everything is paid off? I am projecting to have this paid off within six months, but would this make the process easier?

Any other advice would be greatly appreciated.