Technically, 3-4 weeks is enough for gas to go bad. (but unlikely)
What happens when you jump it off of a car? Does it die after a few minutes or keep going? If you keep it hooked to the car?
The greatest pull on the battery comes when you crank it. I can’t imagine that if you’re able to do that then the batt can’t handle the rest.
But now that I re-read your posts, you say it ‘cranks a little’...Is it turning over and then cutting off or never actually getting going?
Also, those lil’ batteries will run down FAST if you keep trying to crank it over and over without success….