MacBook Pro 13-inch Mid-2013- Unexpected Shutdowns

I have been having this issue for quite sometime now since June. The laptop was previously owned and had the OS wiped. So I installed Catalina on it successfully and it had all the necessary defaults. However, it has been shutting down on me unexpectedly while running apps such as Safari, Zoom, and MS Office, even with a charged battery. I can workaround the problem by keeping it plugged to the charger and close the laptop and open it when it goes into sleep mode (because it apparently doesn't shutdown completely.) There are times where I'll start up the laptop and it'll be deadly slow when I login (rainbow spinning wheel) and launch any app.

Thus far, I've ran the diagnostic tool and it gave me these codes:

• PPP004- 60W power adapter required and it is reading it as a 45W power adapter (I am charging it with a 60W MagSafe power adapter.)
• PPP002- Another related issue with the power adapter
• PPT004- Battery requires service. (I have yet to look into how many cycles it still has.)

I have also reset my SMC several times and have only reset the PRAM once in hopes that it'd adjust the issue. From all this, I think that the issue is likely internal and with the hardware, but I just don't know what specifically is causing it. If anyone has any insights or advice, it'd be greatly appreciated.