@AsperknollsWelcome to HP Community!
I understand that the Spectre 13-4003dx laptop charging light blinks, after changing the charging port.
AS you have stated that you have already replaced the charging port and performed the hard reset.
The issue seems to be the hardware part. I would suggest you contact our phone support. They will help you.
Here is how you can get in touch with phone support.
1)Click on this link - www.hp.com/contacthp/
2)Select the product type.
3)Enter the serial number of your device or select let HP detect your product option.
4)Select the country from the drop-down.
5)You should see the HP phone support number or Chat option listed.
Keep us posted,
If you would like to thank us for our efforts to help you,
Give us a virtual high-five by clicking the 'Thumbs Up' icon below, followed by clicking on the "Accept as solution" on this post,
Have a great day!