Update 2: After some back and forth between me and HostGator (them saying I need a SiteIP and me saying PyAny doesn't provide one that I could find) I told them I wanted to get on the same page and re-explained the situation. I said:
"Ok I want to make sure we're on the same page before we proceed, because reading through the PythonAnywhere forums other people are having the same problem but with GoDaddy. What I would like to do is continue to use HG's name servers, but set up mapping so that engineermyworld.com points to a server with the hostname webapp-XXXXXX.pythonanywhere.com."
I then copied the following text from PyAny staff that I found on this forum page: https://www.pythonanywhere.com/forums/topic/14940/#id_post_77795
"You don't need an IP address or nameservers to point a domain that is registered with them at our servers. (Also, it would be really weird to provide both a nameserver and an IP address to a domain registrar -- you only need one or the other.)
What you need to do is:
Use GoDaddy's nameservers
In their DNS administration pages, create a CNAME record that points www.yourdomain.com at the value we provide on the "Web" page (which will look something like webapp-123456.pythonanywhere.com.
A CNAME record is a way to tell the DNS system where a website is without needing to use an IP address."
-PythonAnywhere staff
HostGator responded with:
"Okay, I will update it as you say.
I have updated the CNAME for you now.
Is there anything else I may assist you with, today?"
I don't know why me quoting PythonAnywhere staff all of a sudden seemed to have helped but what has changed on my cPanel is there is a "Name" of www.engineermyworld.com and "Record" of the webapp-(etc). Some CNAME "Names" that were auto-registered at the time of the domain's purchase are still listed: cpanel.engineermyworld.com., webmail.engineermyworld.com., and ftp.engineermyworld.com. which have a "Record" of engineermyworld.com. HostGator says it should populate in 4-8 hours and I hope that's so. Also, because they added the Name www.engineermyworld.com I changed my web app name from engineermyworld.com to www.engineermyworld.com., which is what the PythonAnywhere staff recommended for another user in the forum I liked above anyway. I'll check back if it works (or) after a reasonable amount of time (4-8 hours or more) to update again.