Just want to post an update here...
I got my site moved last weekend. I accomplished it by following these steps:
1. copy all directories and files over to the new site.
This effectively installed UBBT 7.5.3
Set permissions according to https://www.ubbcentral.com/doc_install.php
2. Create the database manually (I tried using the full backup method from the old site, but the install threw an error on the Create Database step -- invalid name, I think. New site required a specific prefix to the DB name.)
3. Took a UBB threads backup of the old forum, downloaded all the files (one for each table). Then using phpMyAdmin on the new site, imported each table one at a time. This took the longest time in all the process.
4. Changed the DNS registration to point to the new server.
5. Edited includes/config.inc.php and changed the paths where the ip-address was used to use URL addressing.
6. After a few hours waiting for the DNS change to propagate, the new site was up and working.
7. Had to tweak the webcam software, changing a few path designations to upload to the proper locations.
I realize I am still running the old 7.5.3 version. So I'll get going on upgrading to 7.5.6 pretty soon.
I sure appreciate everyone's helping out here, especially Ruben's. I could not
have completed the project without the help.