Spent some more time Googling and searching this morning and after a bit more fiddling, I got it working! One note that I didn't see in the knowledgebase (not sure if it was the "missing link" to make this work or not) was to delete/drop all the existing tables in the new database prior to import. I did not do that the first time, and I wonder if maybe that caused the issues?
Anyway, here were my steps:
1 - Backed up HESK site folder and SQL database.
2 - Upgrade HESK 2.5.5 p5 to 2.6.5 on production site using Klemen's step-by-step instructions (worked great)
3 - Confirmed production site is running fine on new version.
4 - Install HESK 2.6.5 on new server.
5 - Login to new HESK server, confirm it works (probably could've setup settings at this point, chose not to)
6 - Export SQL database from current production site (data and structure).
7 - Using phpMyadmin, open new server SQL database (created via Softaculous script) and drop all tables
8 - Still in phpMyAdmin, import production site SQL tables into new server SQL database
9 - Attempt to log in to new server admin, find password doesn't work

10 - Upload reset_password.php, follow instructions, gain access to HESK on new server with all data!
I probably could've done a cPanel-to-cPanel copy which would've been easier, but there's a lot of other crap on the old server I'm not bringing with me, and I wanted to have a clean start. I'll spend today getting the settings updated, customized email templates copied over, etc., and will test prior to switching everything over.
I'm more stubborn than smart, so I don't know if the above steps are correct (or if there's a much easier way to do this), but it worked for me.
Thanks...
Jason