How to log in using external keyboard on MacBook Pro
You need to replace the MacBook keyboard.

(Or at least disconnect it from the logic board)

The problem is you have two keyboards "battling" it out for what keys get pressed.  The top case assembly which consists of the track pad and the keyboard are both USB devices.  You can have several of these devices plugged in simultaneously - in fact, I have 3 keyboards and one mouse (2 keyboards are for macros) and the iMac happily accepts input from all three simultaneously.

What's happening is you're keyboard is still plugged in and the ⇧ Shift key is acting like it's being pressed because the contacts are shorting out.  I wrote a lengthy answer on this very topic:  Keyboard key stuck or not being recognized; how to fix.  So, even though it's a different keyboard, you're Mac doesn't know; actually it doesn't even care.  It's going to process each key press as if you wanted it that way.

You're only (temp) solution is to disconnect it from the logic board.  There's no amount of resetting the NVRAM and/or the SMC, re-installation of the OS, etc. is going to bypass a physical problem.  

You have a liquid damaged MacBook which means there's probably much more damage - like corrosion of the electrical contacts - that you're going to have to contend with.  Take it in for service!



Below is an image from iFixit.com showing the location of the keyboard connector.  Disconnect this to get your external keyboard working again.  You will need a mouse or another track pad.