Question [Q] Help: Phone Powers Off & Only Switches on Again When Charging

Hi,
hoping someone can give some advice as I'm running out of ideas. Essentially what happens is that my Desire S powers off whilst I am using it, mostly whilst I am navigating through the phone or using any kind of App (not always immediately, but always after a few minutes). I can power it up again afterwards but it will not boot up all the way & instead powers off again. However, if I connect it to a power source it will power up all the way again and it also doesn't power off whilst I use it and it is connected to a cable power source. So far I have:

- Bought a new battery
- Placed folded paper on battery
- Done a Factory reset (everything is stock, phone hasn't been rooted)

Through all of this the problem has continued. Another weird thing that has been happening whilst all this is going on is that the phone seems to have issues keeping the date & time synced. It sets the date to some day in 1980 & time is also incorrect. I have to manually set the date & time as it doesn't correct itself over time.

Any help or suggestions would be much appreciated.