Replaced battery and now the laptop won't stay on without charger

I've got an early 2013 MacBook Pro that had the Service Battery notification and the battery was lasting about 30 minutes. I purchased a replacement battery and the appropriate tools, left the laptop on until the battery was very low on charge (<5%) and then proceeded to replace the battery. It took me awhile, but I didn't run into any issues and I got it all back together.

I plugged in the charger and it went to orange indicating it was charging and I turned the laptop on and it booted without issue, indicating 58% battery life. I then unplugged the charger to make sure the battery was working, and the laptop stayed on for about 10 seconds, then just shut off. I plugged the charger back in and the system booted back up on its own and indicated it was still charging. I double checked the battery connection internally and left it charging overnight.

The charge light is now green and once in the OS the battery indicates it is 100% charged, but I'm still experiencing about 10 seconds of run time after disconnecting the charger. I tried a PRAM and SMC reset, and I believe both were successful but I'm not positive on if the charger should or shouldn't be plugged in for the SMC reset so I've tried both.

coconutBattery indicates the new battery is at 100% charge so I'm kind of at a loss for what's going on here, and would really appreciate any help! I'm comfortable with hardware at this level, but I'm not terribly familiar with Macs on a whole so I'm just wondering if I trashed this thing somehow or where to go from here...