Turns out it was a VPN/IP address issue. I work remotely and when I was logging into my work account to access my server, I have to go through our VPN. This makes the IP address moot, so when I was using the IP address as the "Publisher Folder Hostname", ArcGIS couldn't really resolve that. When I switched to using the device name (aka computer name) as the publisher folder hostname, I was able to publish my service successfully. In this case, my device name was something like "DEKSTOP-XX1234" but it could be pretty much anything. I found it by going to Settings > System > About in Windows 10.
I also switched the server folder path to a local path on the server since I didn't end up using a shared network path.