So after testing last night, what worked was moving the PC/MAC jumper on the relay board to the MAC position, and connecting the new desktop tower via the serial port. I also tested this on the old PC and it still worked. Strangely though, the USB to serial adapter did not work, but I think this has more to do with drivers for that particular cable.
Although the default jumper position for PC RS232 applications is the PC setting, the manual specifies that the relay board won’t be damaged if it is set to MAC:
Input data voltage select, PC +/- 12V RS232, MAC +/5V RS232 specification. Use MAC mode for Mac systems, PC laptops, USB to Serial Adapters, and TTL data sources. It is okay to try both settings if you are unsure for your application. No damage will result due to improper setting of this jumper.
I went down this route after reading this post.
Thank you @Jacob for taking your time to help with this.