MacBook Pro shuts down after login

Some additional information. Attempted to restore from an older copy of Time Machine, which did not work...machine wouldn't boot. So returned to Mac Recovery and reinstalled OS. This time it downloaded and installed Catalina. After install machine worked properly for about 30 minutes, then shut down. Once again it shuts down after typing in my password at the login screen. Still works fine in Safe Mode.

MacBook Pro 15" Retina Mid 2014 16GB Ram, 256GB SSD

More troubleshooting. Hopefully someone can come up with an answer. First of all, a new battery was installed 9 months ago, however, if I unplug the MagSafe connector the computer immediately shuts down. It will not run at all on just the battery. The battery icon shows 'Condition:Normal', 'Battery is Charged', and the status bar shows 100%, but system information under power says Charging No, Fully charged No. With the power adapter connected it will run. After the previous information I posted I did more troubleshooting and discovered that in Safe Mode, the computer would shut down after approximately 2 1/2 hours, but would reboot and run fine. It still would not get past the login screen in normal boot mode. Instead of erasing the SSD and reinstalling the OS, I deleted the volume, erased and reformatted the drive, then reinstalled OS. This time it installed Mavericks (what the machine originally came with), and it boots properly in normal mode. I can log in, and the computer continues to work, at least for an hour so far. So, do I have a MagSafe problem, a battery problem, logic board problem, or something else I'm not seeing? Any help would be appreciated.