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Try removing the batteries, and leaving the camera on a (dry, non-humid, non-dusty) shelf for overnight. Then put the batteries back in and try the camera again.
That will make sure the problem isn't just a temporary glitch that needs a reset... if it doesn't cure the problem.
After that... it's natural that the camera turns On when connected via USB to a computer - that both signals it to switch On, and provides power to it. And it's a good sign that the problem isn't major.
You will probably have to have the camera serviced, if it truly is the button, not the battery, that is at fault. And if the battery is the problem, it would be because it is getting quite old, and has lost the power-delivering capability as all rechargeable batteries of electronic devices do with time.
If over 2 years old, that is quite possible. And if you try a new rechargeable, be sure to charge it up for at least 6 hours, initially, before trying it in the camera.
Let me know, and I will advise you further as appropriate. Thanks.