Car battery: able to jump and charge or just replace?

I have a Valucraft 35-VL battery in my 2002 Toyota Avalon XL. I moved from NC to Western NY and during the first winter, despite turning it on for a few minutes everyday, the battery died and then completely drained. I brought the Stanley Fatmax 1200 AMP Jump starter. Green light game on signaling it was connected, I turned it on, lights came back on in the car and on the dash after a few minutes, but when I go to crank it on it revs for a second like it's starting before it flusters, lights on dash starts blinking, gas level goes down and it just stalls like it's struggling. I let go and turn it back to the idle/off position with my key and Everytime after it just keeps doing that no matter what. The second to last time when I turned it back off it still sounded like a belt or something was trying to move for a few seconds before it stopped. There was some previous blue stuff on the terminals (my mind is blanking on the word right now) that I scrubbed off with a tooth brush. I don't have the tools to get the battery out myself and can't get the car to AutoZone to test/replace the battery. I don't have anyone readily available to help jump my car but if push comes to shove I can hopefully find someone. What should I do? Thanks in advance!